modelling of phase transformation in hot stamping of boron steel

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MODELLING OF PHASE TRANSFORMATION IN HOT STAMPING OF BORON STEEL by Jingqi Cai A thesis submitted for the degree of Doctor of Philosophy of Imperial College London and the Diploma of Imperial College London Department of Mechanical Engineering Imperial College London January 2011

Transcript of modelling of phase transformation in hot stamping of boron steel

Page 1: modelling of phase transformation in hot stamping of boron steel

MODELLING OF PHASE

TRANSFORMATION IN HOT

STAMPING OF BORON STEEL

by

Jingqi Cai

A thesis submitted for the degree of Doctor of Philosophy of Imperial College

London and the Diploma of Imperial College London

Department of Mechanical Engineering

Imperial College London

January 2011

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Abstract

Knowledge of phase transformations in a hot stamping and cold die quenching

process (HSCDQ) is critical for determining physical and mechanical properties of

formed parts. Currently, no modelling technique is available to describe the entire

process. The research work described in this thesis deals with the modelling of phase

transformation in HSCDQ of boron steel, providing a scientific understanding of the

process. Material models in a form of unified constitutive equations are presented.

Heat treatment tests were performed to study the austenitization of boron steel.

Strain-temperature curves, measured using a dilatometer, were used to analyse the

evolution of austenite. It was found that the evolution of austenite is controlled by:

diffusion coefficient, temperature, heating rate and current volume proportion of

austenite. An austenitization model is proposed to describe the relationship between

time, temperature, heating rate and austenitization, in continuous heating processes. It

can predict the start and completion temperatures, evolution of strain and the amount

of austenite during austenitization.

Bainite transformation with strain effect was studied by introducing pre-deformation

in the austenite state. The start and finish temperatures of bainite transformation at

different cooling rates were measured from strain-temperature curves, obtained using a

dilatometer. It was found that pre-deformation promotes bainite transformation. A

bainite transformation model is proposed to describe the effects of strain and strain rate,

of pre-deformation, on the evolution of bainite transformation. An energy factor, as a

function of normalised dislocation density, is introduced into the model to rationalise the

strain effect.

Viscoplastic behaviour of boron steel was studied by analyzing stress-strain curves

obtained from uni-axial tensile tests. A viscoplastic-damage model has been developed

to describe the evolution of plastic strain, isotropic hardening, normalised dislocation

density and damage factor of the steel, when forming in a temperature range of 600°C

to 800°C.

Formability tests were conducted and the results were used to validate the

viscoplastic-damage model and bainite transformation model. Finite element analysis

was carried out to simulate the formability tests using the commercial software,

ABAQUS. The material models were integrated with ABAQUS using VUMAT. A good

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agreement was obtained between the experimental and FE results for: deformation

degree, thickness distribution, and microstructural evolution.

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Acknowledgements

I am most grateful to my supervisor, Prof. J. Lin, for his guidance, support,

encouragement and supervision given throughout this academic program, which

helped me greatly in the understanding of the project and writing of the thesis.

I would like to acknowledge Dr. J. Wilsius (ArcelorMittal, France) regarding the

supply of materials, the guidance on this work, and the financial and academic support

on this project from ArcelorMittal, France.

My heartfelt gratitude goes out to Prof. T. A. Dean of the University of Birmingham,

for his valuable advice on both my work and thesis.

Very many thanks to Dr. B. Roebuck, Dr. K. Mingard, and Mr. A. Falshaw (National

Physical Laboratory, UK) for their help on the microhardness tests.

For the friendly cooperation and useful discussions throughout the period of

research work on this project, I wish to express my appreciation to many past and

present colleagues in both University of Birmingham and Imperial College London, in

particular to: Dr. J. Cao, Dr. A. Foster, Dr. Y. Lin, Ms. N. Li, Ms. Q. Bai, Mr. M.

Karimpour, Dr. S. Wang, Dr. Z. Shi. Huge thanks to all who supported me in any

respect during the completion of the project.

I wish to present my deep sense of appreciation to my husband, L. Wang, for his

support, his help, his tolerance, his encouragement, and his love.

Last but not least, big thanks to my parents, P. Cai and L. Han, and my sister, M,

Cai, thank you for always believing in me.

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Nomenclature

n Work hardening exponent of a material

sM The start temperature of martensite transformation

G Gibbs free energy

areaA Surface area

Interfacial free energy

sG Strain energy

VG Volume free energy release

A

VG , M

VG Volume free energies of austenite and martensite, respectively

MG Volume free energy difference between austenite and martensite at sM

MG Volume free energy difference between martensite and austenite

eqT Equilibrium temperature of austenite/martensite transformation

sA Starting temperature of austenite transformation from martensite

V Volume of the nucleus

dG Dislocation interaction energy

Poisson ratio

s Shear strain of the nucleus

Shear modulus of the austenite

a, c Diameter and thickness of a martensite nucleus, respectively

sB Bainite start temperature

max

Bv Maximum volume fraction of bainitic ferrite

B

norv Normalised volume fraction of bainitic ferrite

Bv Actual volume fraction of bainitic ferrite

w Widmanstätten ferrite

wsI Widmanstätten ferrite nucleation rate

wsT Start temperature of Widmanstätten ferrite transformation

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B

mG The maximum possible free energy change on nucleation

0

B

mG Initial value of B

mG

B

NG The value of universal curve representing the minimum necessary free

energy change for displacive nucleation of ferrite at the wsT temperature

1 , 2 , Empirical material constants

1K , 2K , 3K ,1K Material constants

L The parameter related to austenite grain size

1 , 2 Empirical material constants

Cx Mean carbon concentration of the material during bainite transformation

B

avru Average volume of a single bainite subunit (mm3)

B

oI Initial value of BI

B Empirical autocatalysis constant

1 , 2 , 3 Material constants

i wt% of element i in solid solution

bK A material constant

b

NQ Active energy of bainite nucleation

b

c Critical curvature of a growing phase at the growth tip which gives the

maximum growth rate

0

bv growth rate of bainite

bD Diffusivity coefficient in bainite formation

0

b Carbon supersaturation

b

eV Volume filled by the grains of the new phase without impingement,

0

bV Initial volume to be transformed

b Multiplying factor adjusting the actual lath velocity

1

bK A material constant

2

bK A ratio accounting for austenite grain boundary diffusion during bainite

nucleation

cQ Activation energy for bulk diffusion of carbon in austenite

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rG Total free energy change of when a new particle formed

MECHG Energy by mechanical work

N Normal stress on the habit plane

The component of the shear stress on the habit plane

, Dilatational and shear components of the shape deformation,

respectively

if Material constants employed in bainite transformation model

*T Temperature corresponding to the shortest incubation time

fx Bainite fraction

1 , 2 Material constants employed in bainite transformation model

, , Austenite, cementite and ferrite phases in steel

C

Carbon concentration in austenite at the / interface

C

Carbon concentration in austenite at the / interface

C

Carbon concentration in ferrite at the / interface

C0 Average carbon concentration in the material

r0 Initial radius of the cementite particle or initial interface position

r Instantaneous position of the / interface

r Instantaneous position of the / interface

D , D , D Carbon diffusion coefficients in austenite, cementite, and ferrite,

respectively

f Transformed volume fraction of austenite

1A, 2A

Temperature dependent parameters

0I, 0g

Constants

G Energy barrier for stable austenite nucleation

VQ Activation energy of austenite grain growth

N Factor that mainly depends on the morphology of new phase

, Thermal expansion coefficient of ferrite and bainite

x Incubation coefficient

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va Fraction of austenite

εθ Thermal strain change during phase transformation

T Temperature

D Temperature dependent parameters

D0 Material constant

QD Material constant

R Universal gas constant

A Material constants

Tc Critical temperature of austenite phase transformation

ma, na Material constants

γ1, γ2 Rregulation of heating rate in the phase transformation

α, β Thermal expansion coefficient of ferrite and austenite, respectively

θ Material constant related to volume difference between original material

lattice structures and the austenite lattice

s Engineering stress

F Tensile force

e Engineering strain

L0 Initial gage length

△L Change in gage length

E Young‟s modulus

elateral, eaxial Lateral contraction (strain) and the axial strain of a tension test

eT Total strain

ee, ep Elastic and plastic engineering strain, respectively

%L Percent elongation

Lf Gage length when the failure occurs

A0, Af Initial cross-sectional area and the cross-sectional area at fracture

At Cross-sectional area

Lt Gage length of a tensile sample

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Stress

F Force

Strain

w0, w Initial width and the instantaneous measure of width of the gage section

D0, D Initial diameter and the instantaneous measure of diameter of the gage

section, respectively

r Anisotropy property

r Normal anisotropy

r Planar anisotropy

T Total strain

e Elastic strain

p Plastic strain

p Plastic strain rate

Stress rate

Tm Melting temperature of a metal

E0 Material constant

QE Active energy related material constant

* ,*N Coefficients which are characteristics of each material and temperature

*K Temperature dependent material constant

*

1m ,*

2m Temperature dependent material constants

*

1M ,*

2M Temperature dependent material constants

e Equivalent stress

H Internal stress due to isotropic hardening

k Initial yield stress

nc Material constant

L Dislocation slip length

Dislocation density

Normalised dislocation density

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b Burger‟s vector

M* Material constant defining the texture of the polycrystalline structure

*

1k , *

2k Material constants

dynamic , static Dynamic and static recovery of normalized dislocation density

*M Mobility of dislocation recovery

* Average energy per unit length of dislocation

*

i , i Material constants

C Temperature dependent material constant

K0,k0,n0,B0,C0 Material constant

Q Active energy parameter

QC Active energy for static recovery

Qn Active energy related material constant

m Mean normal stress

y Parameter related to the material flow stress

p

e Equivalent plastic strain rate

cgD Damage due to continuum cavity growth

voidr Void radius

voidl The spacing of growing voids

nd Material constant

m The minimum creep rate

d Temperature dependent material constant

0 Material constant

QD Material constant related to active energy of damage

nett Nett stress

T Undercooling of bainite transformation

eqT Equilibrium temperature of austenite and bainite

A

VG , B

VG Free energies of initial (austenite) and final (bainite) states, respectively

QV Latent heat of fusion per unit volume

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*r Critical radius of new formed phase

*G Maximum excess free energy

G Total free energy change of bainite transformation

A1, A2, A3 Material constants

B1, B2 Material constants

n1, n2, n3, n4 Material constants

XB, XM Volume fraction for the transformation from austenite to bainite and

martensite, respectively

T1 Temperature when the maximum growth rate of bainite achieved

T2 Temperature when the maximum nucleation rate of bainite achieved

B Material constant

f Energy factor due to the accumulation of dislocation

d0 Initial diameter of the central hole of formability test sample

dt, Diameter of the central hole at time t

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Contents

ABSTRACT ............................................................................................................. 2

ACKNOWLEDGEMENTS ............................................................................................. 4

NOMENCLATURE ...................................................................................................... 5

CONTENTS ........................................................................................................... 12

LIST OF FIGURES ..................................................................................................... 16

LIST OF TABLES ....................................................................................................... 21

CHAPTER 1 INTRODUCTION .................................................................................. 22

1.1 THE MODERN AUTOMOTIVE INDUSTRY ....................................................................... 22

1.2 ENVIRONMENTAL IMPACT ........................................................................................ 23

1.3 SAFETY REQUIREMENT ............................................................................................ 24

1.4 ADVANCED DESIGN AND MANUFACTURING TECHNIQUE ................................................. 25

1.4.1 Light weight structures ................................................................................. 25

1.4.1.1 Frame structures ............................................................................................. 26

1.4.1.2 Shell structures ............................................................................................... 28

1.4.2 Light weight materials .................................................................................. 30

1.4.2.1 Advanced high strength steels (AHSS) ............................................................ 30

1.4.2.2 Aluminium and magnesium ............................................................................ 33

1.5 CHALLENGES AND OBJECTIVES OF THE RESEARCH .......................................................... 34

1.6 STRUCTURE OF THESIS ............................................................................................. 36

CHAPTER 2 ADVANCED TECHNOLOGIES IN FORMING HIGH STRENGTH PANELS FOR

AUTOMOBILES ........................................................................................................ 38

2.1 INTRODUCTION ..................................................................................................... 38

2.2 TRADITIONAL FORMING TECHNIQUES ......................................................................... 38

2.3 HOT STAMPING AND COLD DIE QUENCHING ................................................................. 42

2.3.1 Coated boron steel ........................................................................................ 42

2.3.2 Forming process ............................................................................................ 43

2.3.3 Applications and market trends .................................................................... 45

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2.4 CHALLENGES OF HOT STAMPING AND COLD DIE QUENCHING PROCESS ............................... 48

CHAPTER 3 PHASE TRANSFORMATION AND MODELLING METHODS ...................... 49

3.1 INTRODUCTION ..................................................................................................... 49

3.2 PHASE TRANSFORMATION MECHANISMS ..................................................................... 49

3.2.1 Austenitization .............................................................................................. 49

3.2.1.1 Nucleation ....................................................................................................... 49

3.2.1.2 Grain growth of austenite ............................................................................... 52

3.2.1.3 Homogenization of austenite.......................................................................... 52

3.2.2 Bainite transformation.................................................................................. 53

3.3 MODELLING OF PHASE TRANSFORMATIONS ................................................................. 55

3.3.1 Modelling of austenitization ......................................................................... 56

3.3.1.1 Kinetic model .................................................................................................. 56

3.3.1.2 Modelling of austenitization based on Fick’s second law ............................... 57

3.3.2 Modelling of bainite transformation ............................................................ 62

3.3.2.1 Nucleation controlled model .......................................................................... 62

3.3.2.2 Diffusion controlled model ............................................................................. 65

3.3.2.3 Modelling of strain effect on bainite transformation ..................................... 68

CHAPTER 4 MODELLING OF AUSTENITIZATION ...................................................... 71

4.1 INTRODUCTION ..................................................................................................... 71

4.2 EXPERIMENTAL INVESTIGATIONS ............................................................................... 71

4.2.1 Experimental program .................................................................................. 72

4.2.2 Facilities and phase transformation measurement ...................................... 73

4.3 MODELLING OF AUSTENITIZATION ............................................................................. 76

4.3.1 Modelling of incubation time ........................................................................ 76

4.3.2 Modelling of the evolution of the austenite fraction .................................... 77

4.3.3 Evaluation of dilatometry strain ................................................................... 78

4.3.4 The unified constitutive equations ................................................................ 78

4.4 DETERMINATION AND VALIDATION OF THE UNIFIED MATERIAL MODEL FOR AUSTENITIZATION 79

4.5 SUMMARY ............................................................................................................ 82

CHAPTER 5 THERMO-MECHANICAL TESTING ......................................................... 84

5.1 INTRODUCTION ..................................................................................................... 84

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5.2 UNIAXIAL HOT TENSILE TESTS .................................................................................... 84

5.2.1 Aim ................................................................................................................ 84

5.2.2 Equipment and testing procedure ................................................................ 85

5.2.3 Test-piece design .......................................................................................... 86

5.2.4 Temperature distribution in test-pieces of different gauge length .............. 87

5.2.5 Anisotropy at hot stamping conditions ......................................................... 90

5.2.5.1 Definition of anisotropy .................................................................................. 90

5.2.5.2 Anisotropy of the tested material ................................................................... 91

5.2.6 Mechanical behaviour................................................................................... 94

5.2.6.1 Effect of austenitization conditions ................................................................ 94

5.2.6.2 Mechanical characterisation ........................................................................... 96

5.3 FORMABILITY TESTS ................................................................................................ 98

5.3.1 Equipment and testing procedure ................................................................ 98

5.3.2 Test-piece design .......................................................................................... 99

5.3.3 Mechanical response of the material ......................................................... 100

CHAPTER 6 DEVELOPMENT OF UNIFIED VISCOPLASTIC DAMAGE CONSTITUTIVE

EQUATIONS ......................................................................................................... 104

6.1 INTRODUCTION ................................................................................................... 104

6.2 DEVELOPMENT OF VISCOPLASTIC CONSTITUTIVE EQUATIONS ......................................... 104

6.2.1 Flow rule ...................................................................................................... 105

6.2.2 Dislocation density evolution ...................................................................... 108

6.2.3 Isotropic work hardening ............................................................................ 111

6.2.4 Uniaxial unified viscoplastic constitutive equations ................................... 112

6.3 DEVELOPMENT OF VISCOPLASTIC DAMAGE EQUATIONS ................................................ 113

6.3.1 Damage evolution ....................................................................................... 114

6.3.2 Modification of the material flow stress rule ............................................. 117

6.4 UNIAXIAL VISCOPLASTIC DAMAGE CONSTITUTIVE EQUATIONS ........................................ 117

6.5 DETERMINATION OF THE EQUATIONS VIA OPTIMISATION .............................................. 118

6.6 SUMMARY .......................................................................................................... 121

CHAPTER 7 INVESTIGATION OF PHASE TRANSFORMATION DURING COOLING ..... 122

7.1 INTRODUCTION ................................................................................................... 122

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7.2 EXPERIMENTAL INVESTIGATION OF PHASE TRANSFORMATION UNDER COOLING PROCESSES .. 122

7.2.1 Phase transformation without pre-deformation ........................................ 122

7.2.2 Bainite and martensite transformation of pre-deformed material ............ 126

7.3 MODELLING OF PHASE TRANSFORMATION UNDER COOLING PROCESSES .......................... 130

7.3.1 Modelling of bainite transformation without pre-deformation ................. 130

7.3.1.1 Nucleation and growth ................................................................................. 130

7.3.1.2 Evolution of volume fraction ........................................................................ 134

7.3.1.3 Unified constitutive equations ...................................................................... 135

7.3.1.4 Calibration of the model ............................................................................... 136

7.3.2 Modelling of bainite transformation of pre-deformed material ................ 138

7.3.2.1 Strain effect ................................................................................................... 138

7.4 VALIDATION OF THE MATERIAL MODELS .................................................................... 142

7.4.1 Implementation of the material models ..................................................... 142

7.4.2 Simulation results ....................................................................................... 142

CHAPTER 8 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH ............... 148

8.1 CONCLUSIONS ..................................................................................................... 148

8.2 SUGGESTED FUTURE WORK .................................................................................... 151

REFERENCES ......................................................................................................... 153

APPENDIX ......................................................................................................... 167

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List of figures

Figure 1.1 Emissions from the life cycle of a typical vehicle [6]. .................................. 24

Figure 1.2 Tubular frame structure of BMW C1. .......................................................... 27

Figure 1.3 Profile based frame structure of the Ferrari F360 Modena. ........................ 27

Figure 1.4 Curved profile extrusion (rounding during extrusion) of a variably curved

bumper (left) compared with the stretch bent bumper (right). ...................................... 28

Figure 1.5 High strength steel is employed in B-pillar of VW Passat. .......................... 28

Figure 1.6 A side panel made of tailored blanks.......................................................... 29

Figure 1.7 Schematic comparison among AHSS steels (TRIP, DP, CP, MS), low

strength steels (IF MILD, IS) and traditional HSS steels (IF-HS, BH, CMn, HSLA). ..... 31

Figure 1.8 Audi AL2 with an all aluminium body structure. .......................................... 33

Figure 2.1 Comparison of maximum bending angle of MS 1200 with (right)/ without (left)

laser-assisted bending [57]. ........................................................................................ 40

Figure 2.2 Hydroformed Engine Cradle made from welded DP 280/600 tube. Draw

bending, Bending Angle>90°[32]. ............................................................................... 41

Figure 2.3 CCT diagram of USIBOR 1500 P provided by ArcelorMittal. ...................... 43

Figure 2.4 Temperature profile and illustration of the HSCDQ process. Microstructures

at different stages are indicated. ................................................................................. 44

Figure 2.5 Potential applications of hot stamped laser welded blanks combining

USIBOR 1500 P and DUCTIBOR 500 P [43]. ............................................................. 46

Figure 2.6 Comparison of yield strength and ultimate tensile strength of different types

of hot stamped steel, HSS steels and other types of AHSS steels. ............................. 47

Figure 3.1 Schematic diagram of an isothermal TTT diagram of the formation of

austenite. a) Eutectoid steel (C% = 0.8); b) Hypoeutectic steel (C% < 0.8). ................ 51

Figure 3.2 The preferred nucleation sites for austenite formation from the parent phase

of a) pure ferrite, b) ferrite with spheroidized cementite and c) lamellar pearlite. ......... 51

Figure 3.3 a) Illustration of austenite nucleation, grain growth and homogenization in a

two-phase steel; b) Carbon distribution profile in each phase at various stages

of austenitization. ........................................................................................................ 53

Figure 3.4 Schematic illustration of the development of upper and lower bainite in

steels [90]. .................................................................................................................. 54

Figure 3.5 Transmission electron micrographs illustrating a) upper [91] and b) lower

bainite [92]. ................................................................................................................. 55

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Figure 3.6 Illustration of austenitization for a pearlite and ferrite parent material. a)

Parent phase; b) Grain growth of austenite; c) Carbon concentration profile when t=0.

................................................................................................................................... 61

Figure 3.7 Modelling of carbon distribution in austenite, employing the model in [71]

based on Fick‟s second law. ....................................................................................... 62

Figure 4.1 Test program and temperature profile for phase transformation study. ...... 72

Figure 4.2 Design of samples for heat treatment tests. ............................................... 73

Figure 4.3 A typical strain-temperature curve obtained from heat treatment test. ........ 74

Figure 4.4 Experimental start and finish temperatures for austenitization of the boron

steel, and volume fraction of austenite at heating rates of 1.25℃/s and 5℃/s. ............ 74

Figure 4.5 Strain-temperature curves within a temperature range from 727℃ (1000K) to

950℃ (1223K) for tests under heating rates of 5℃/s, 10℃/s and 30℃/s, respectively. 75

Figure 4.6 Test program and temperature profile for evaluation of austenite volume

fraction at different temperature levels. ....................................................................... 76

Figure 4.7 Comparison of experimental (symbols) and computed (solid curves) thermal

strain-temperature relationships for the phase transformation parts from dilatometry

curves. Heating rate varies from 5℃/s to 30℃/s. ......................................................... 80

Figure 4.8 Comparison of experimental (by triangles) and computed (by solid lines and

starts) starting and finishing temperatures of austenitization process, as well as the

evolution of fraction of austenite under heating rates of 1.25℃/s and 5℃/s. ................ 81

Figure 4.9 Evolution of experimental (symbols) and computed (solid curves) fraction of

austenite vs. temperature. Heating rates are 1.25℃/s, 5 ℃/s and 30℃/s. ................... 82

Figure 5.1 Gleeble 3800 system used for hot-tension. (a) The control console with a

Digital Closed-Loop Control System and the load unit with test chamber are shown in

the picture. (b) Gleeble test unit showing attached thermocouple wires and C-Gauge.85

Figure 5.2 Temperature profile and testing procedure of hot tension tests. ................. 86

Figure 5.3 A tensile test-piece showing the gauge length, effective gauge length and

diameter. .................................................................................................................... 87

Figure 5.4 Dimensions of tensile test-pieces for hot tensile tests. Gauge lengths (L0),

14mm, 30mm and 50mm. ........................................................................................... 88

Figure 5.5 Schematic illustration of testing procedure. ................................................ 89

Figure 5.6 a) Locations of thermocouples welded on test-pieces. b) Temperature

distribution along test-pieces of gauge lengths 14, 30 and 50mm. .............................. 89

Figure 5.7 Hot zones in test-pieces of different length. ............................................... 90

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Figure 5.8 Average normal anisotropy (without being austenitized) sensitivity to forming

temperature and strain rate (test-pieces were cooled at 50K/s after austenitization)

[145]. .......................................................................................................................... 91

Figure 5.9 Samples were cut along the directions of 0°, 45° and 90° to the rolling

direction. ..................................................................................................................... 92

Figure 5.10 Test programme and temperature profile. ................................................ 92

Figure 5.11 Stress-strain relationships for the samples along various directions (0°, 45°

and 90°) and formed at (a) 600℃ and (b) 800℃.......................................................... 93

Figure 5.12 Test programme and temperature profile. ................................................ 94

Figure 5.13 Stresses at a strain level of 0.1 for test-pieces subjected to different

austenitization conditions (870℃ for 1min, 925℃ for 5 min, and 950℃ for 10 min). .... 95

Figure 5.14 Strain to failure of test-pieces subjected to different austenitization

conditions (870℃ for 1min, 925℃ for 5min, and 950℃ for 10min). .............................. 96

Figure 5.15 Microstructures of test-pieces after tension. Soaking conditions were 870℃

for 1min, 925℃ for 5min, and 950℃ for 10min. ........................................................... 96

Figure 5.16 a) Strain to failure and b) maximum flow stress versus temperature for hot

tension tests at different strain rates (0.01/s, 0.1/s, 1.0/s and 10.0/s). ......................... 97

Figure 5.17 a) Apparatus of formability tests: 1- heating furnace, 2- high speed camera,

3- forming die set, 4- 25 tonne ESH high speed press. b) A 3-D drawing of the die set.

................................................................................................................................... 99

Figure 5.18 Temperature profile and testing procedure of formability test. .................. 99

Figure 5.19 Test-piece dimensions for formability test. ............................................. 100

Figure 5.20 Hot stamped test-pieces. Ram speeds are a) 15 mm/s, b) 100 mm/s. ... 101

Figure 5.21 Normalized diameter dt/d0 vs. ram stoke st. Ram speeds are 15mm/s and

100mm/s. .................................................................................................................. 102

Figure 5.22 Sectional views of deformed regions which were cut from test-pieces. Ram

speeds are a) 15mm/s and b) 100mm/s. ................................................................... 102

Figure 5.23 Thickness distributions against X-Position. ............................................ 103

Figure 6.1 Representation of the three basic tests in the space of the variables p and

p (after Lemaitre et al. [151]). ................................................................................ 106

Figure 6.2 Schematic illustration of decomposition of stress dissipation. .................. 108

Figure 6.3 Schematic illustration of damage mechanism under hot metal forming

condition, after Lin et al. [184]. .................................................................................. 115

Figure 6.4 Comparison of experimental (symbols) and computed (solid curves) stress

strain curves. a) Forming temperature is 700℃, strain rate is from 0.01/s to10/s; b)

Forming temperature is from 600℃ to 800℃, strain rate is 0.1/s. .............................. 120

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Figure 7.1 Test program and temperature profile for phase transformation study. .... 123

Figure 7.2 Dilation curves obtained for phase transformation tests presented in Figure

7.1, when cooling rates varies from 10℃/s to 50℃/s. ................................................ 124

Figure 7.3 Tested starting of bainite and martensite transformation temperatures under

cooling rates of 10, 20, 25, 30 and 50℃/s. ................................................................ 125

Figure 7.4 SEM photographs of the boron steel after the heat treatment test described

in Figure 7.1. Cooling rates of the listed microstructures are 10℃/s, 20℃/s and 25 ℃/s.

Transformed bainite is outlined in the photographs. .................................................. 126

Figure 7.5 Test program and temperature profile for phase transformation study when

pre-deformation was introduced. ............................................................................... 127

Figure 7.6 Critical temperatures of bainite and martensite transformation when pre-

deformation was introduced. ..................................................................................... 128

Figure 7.7 Microstructures for the material after pre-deformation a) at 800℃ and b) at

600℃, to a strain of 0.1 at 1.0/s. Cooling rates are a) 15℃/s and b) 25 ℃/s. Some

bainite is outlined by dashed circles. ......................................................................... 129

Figure 7.8 Changes of volume free energy of bainite/austenite transformation as a

function of temperature [134]. ................................................................................... 131

Figure 7.9 Change of free energy as a function of nucleus size. ............................... 132

Figure 7.10 Growth rate of the size of bainite as a function of temperature. .............. 134

Figure 7.11 Nucleation rate of bainite as a function of temperature. .......................... 135

Figure 7.12 Comparison of predicted starting temperature of bainite transformation and

CCT diagram of boron steel. ..................................................................................... 137

Figure 7.13 Comparison of predicted bainite fraction and experimentally observed

bainite fraction. ......................................................................................................... 138

Figure 7.14 Comparison of experimentally obtained CCT diagram of boron steel with

strain effect (dots) and the predicted data (solid curves and the thick dashed curve) by

the proposed model. a) pre-deformation is at 800℃, b) pre-deformation is at 600℃. 140

Figure 7.15 Comparison of predicted bainite fraction and experimentally observed

bainite fraction, with strain effects. The temperatures of pre-deforming are 600℃ and

800℃, respectively. The strain rate is 1/s, strain is 0.1. ............................................. 141

Figure7.16 a) FE model of formability test built by ABAQUS. b) FE analysis of formed

test piece when the ram speed is b) 100 mm/s and c) 15 mm/s. ............................... 143

Figure 7.17 Heat transfer coefficient with functions of pressure and clearance. ........ 144

Figure 7.18 Comparison of half-cross-section profiles between the experimental result

and FE analysis. Ram speeds are a) 100 mm/s and b)15 mm/s. .............................. 144

Figure 7.19 The predicted and experimentally measured thickness distribution. ....... 145

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Figure 7.20 Comparison of the evolution of central hole diameters between the

experimental result and FE analysis. Ram speeds are 100 mm/s and 15 mm/s. ....... 145

Figure 7.21 Comparison of predicted martensite distribution and observed bainite

distribution. Ram speeds of tests are a) 100 mm/s and b) 15 mm/s. Some bainite is

highlighted by dashed circles. ................................................................................... 146

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List of tables

Table 2.1 Chemical composition of USIBOR 1500 P (Maximum %). ........................... 43

Table 2.2 Growth of hot stamping technology in Europe [43]. ..................................... 47

Table 4.1 Material constants determined for Equation set 4.6. .................................... 81

Table 5.1 Strain rates and temperatures of tension tests at different temperatures. .... 86

Table 5.2 Soaking time and temperatures of tests. ..................................................... 94

Table 6.1 Material constants determined for Equation set 6.32 and 6.33. ................. 121

Table 7.1 Volume fraction of bainite transformation corresponding to the heat treatment

test in Figure 7.1. ...................................................................................................... 126

Table 7.2 Critical phase transformation temperatures in Figure 7.1. ......................... 128

Table 7.3 Constants determined by fitting the predicted starting temperature of bainite

transformation with CCT diagram. ............................................................................ 137

Table 7.4 Constants determined by fitting the predicted fraction of bainite

transformation with experimentally measured fraction of bainite transformation. ....... 138

Table 7.5 Determined material constants for equations of *r and r . ........................ 141

Table 7.6 Determined constants for the equation for X . ........................................... 142

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Chapter 1

Introduction

1.1 The modern automotive industry

The establishment of motor vehicle manufacturing as a distinct industry occurred in

the period between 1889 and 1908. After World War Ⅱ, motor vehicle output expanded

rapidly, reaching 10.5 million units in 1950, 16.3 million in 1960, 30 million in 1969, and

39.1 million in 1973 [1]. The automotive industry is substantially affected by oil price,

economic circumstance, and the development of technology over the world, but is

generally in a steady growth. Its production has grown from small companies

manufacturing simple carriages to the international corporations that now mass-

produce advanced and reliable automobiles across the globe.

Efficiency was strongly concerned in the past which contributed to the sharp

increase in the amount of automobiles. As a consequence, environmental problems

have become more severe, such as the air pollution, greenhouse effect, as well as the

over consumption of gasoline and diesel. Many governments have made strong

commitments to control the negative impact on the environment from automobiles.

Consequently, in recent years, the main objective of the automotive industry is no

longer constrained to the production of a huge quantity of products with high efficiency,

but also targets new approaches to reduce fuel consumption and exhaust emissions.

Minimizing the negative impact on the environment and cost, has accompanied

increases in the safety, appearance and precision. Innovations are needed to met the

emerging energy saving and air-quality demands. Several new technologies were

therefore introduced, such as improved diesel engines, catalytic converters, electronic

fuel injection, turbochargers, high-strength and advanced high strength steels (HSS

and AHSS), aerodynamic bodies and front-wheel drive, etc [2].

In general, in terms of car body manufacturing, vehicles are in a trend of using

improved or new materials which can reduce car body weight, provide fuel economy,

allow smoother surfaces and more complex shapes, and possess high formability and

excellent crashworthiness.

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1.2 Environmental impact

The latest scientific evidence indicates that the accumulations of greenhouse gases

in the atmosphere are altering the Earth‟s climate patterns at unnatural rates. The

Intergovernmental Panel on Climate Change (IPCC) has claimed that global average

temperatures rose by 0.6 ℃ over the last century, and that “most of the warming

observed over the last 50 years is attributable to human activities” [3]. A changing

climate may lead to unpredictable and costly consequences, such as rising sea level,

spreading of infectious diseases, and altered incidence and location of extreme

weather events, including hurricanes, monsoons, and droughts. Although uncertainty

remains regarding climate trends, the threat of climate change has prompted concerns

and actions at national and global levels.

Since the automotive sector is a major source of CO2 emissions, and accounts for

30 percent of CO2 emissions in the industrialized economies of the OECD

(Organization for Economic Cooperation and Development) and about 20 percent

worldwide, efforts to prevent climate change have already led to policies in several of

the world‟s major automotive markets to reduce vehicle-related CO2 emissions. It thus

pushed automotive Original Equipment Manufacturers (OEMs) to produce vehicles

which emit fewer CO2 per kilometre (km) travelled. For example, the Clean Air Act of

1990 in U.S. tightened federal exhaust emission standards for both automobiles and

heavy-duty trucks, and stipulated that the average mileage for all cars be 11.7 km/litre

(27.5 mi/gal) by 1985 [4]. The European Union and Japan have both made strong

commitments to lower the CO2 emissions rates of vehicles. However, debate over

federal Corporate Average Fuel Economy (CAFE) standards continues, while a 2002

California law seeks to regulate vehicles CO2 emissions for the first time. The most

conspicuous policy step is the Kyoto Protocol which aims at a long-term process of

reducing global greenhouse gas (GHG) emissions. As an indicator of growing pressure

in this area, by October 2003, 111 countries had ratified the Protocol, including those

responsible for 60 percent of 2002 global vehicles sales [5, 6].

While CO2 emissions arise at nearly every stage of a motor vehicle‟s life, including

extraction of raw materials and manufacturing of component parts, it is the combustion

of gasoline and diesel fuels that accounts for the greatest share of vehicle –related CO2

emissions (Figure 1.1) [6]. Hence, approaches to reduce automotive industry‟s

negative effects on environment should focus on the control of exhaust emissions,

mainly in CO2, of cars.

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Figure 1.1 Emissions from the life cycle of a typical vehicle [6].

As it has been mentioned above, the fast development of the automotive industry

has worsened the environmental situation; contrarily, emerging carbon constraints

could significantly impact the automotive industry, primarily through pressure to

increase the fuel economy or lower the carbon dioxide (CO2) emissions. Automotive

companies and corporations, as well as material suppliers are thus required to respond

to these pressures.

Among the many measures to decrease fuel consumption and emissions, reducing

weight is proved to be extraordinary effective. A rule of thumb is that a 10% weight

reduction leads to 3%~7% decrease in gasoline consumption and consequently less

emissions [7, 8]. Therefore, car manufacturers and steel plants are doubling their

efforts in searching for all possible approaches to reduce the vehicle body weight.

Using light structures and light materials are the most direct and promising methods.

1.3 Safety requirement

Automobile safety became an issue of increased public concern since the 1960s,

and is a critical factor directing the automotive industry. It is reported by the World

Health Organization (WHO) that auto collisions are the leading cause of injury-related

deaths, which is 25% of the total from all causes. Worldwide, an estimated 1.2 million

people are killed in road crashes each year and as many as 50 million are injured. In

economic terms, the cost of road crash injuries is estimated at roughly 1% of gross

national product (GNP) in low-income countries, 1.5% in middle-income countries and

2% in high-income countries [9]. Direct economic cost of global road crashes have

been estimated at 518 billion dollars [10]. Projections indicate that these figures will

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increase by about 65% over the next 20 years unless there is new commitment to

prevention. Road crashes not only place a heavy burden on national and regional

economies, but also on households whose family members were injured or lost from

road accidents. Therefore, well-tested, cost-effective and publically-acceptable

solutions to the problems are urgently called for.

A study in the United Kingdom concluded that improved vehicle crash protection

(also known as “secondary safety” or “passive safety”) for car occupants and

pedestrians would have the greatest effect in reducing road casualties in Great Britain

[11]. Similarly, in the European Union, there were several directives on frontal and side

impact protection, and tightened regulations on crash tests from the European New Car

Assessment Programme (EuroNCAP) [9]. Globally, significant steps toward improved

protection of occupants of cars were made in 1990s. The concept of “crashworthiness”,

termed as the ability of a structure to protect its occupants during an impact, in vehicle

design, was thus introduced and is now well understood and incorporated into current

car design in highly-motorized countries. Due to the increasing concerns on safety, it is

required that motor vehicles be designed for crashworthiness to protect occupants.

One of the most straightforward ways is increasing the strength of safety components.

Thus exploring and developing new types of high strength materials has become

popular worldwide.

1.4 Advanced design and manufacturing technique

Considering both of the environmental pressures and the increasing safety

requirements, advanced design and manufacturing techniques to strengthen car safety

components and save the mass meantime, are highly demanded by the international

automotive industry. Based on the practice experience, this can be best achieved from

the aspects of both body structure and material. Hence, light weight structures and high

stiffness-mass ratio materials draw great attention from auto makers, and material

suppliers. The employment of ultra high strength components has dramatically

increased in the past two decades, using different types and classes of materials, both

metallic and non-metallic. Furthermore, light weight structures, such as frame

structures and shell structures, are widely used.

1.4.1 Light weight structures

Light weight constructions are optimal if a material is used in component areas

bearing stress and if the material employed is loaded near the yield stress. Such a

structure is primarily designed for strength where high strength materials are used, e.g.

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680MPa aluminium alloy EN-AW7449 in the wing up side of the Air-bus A380, and the

automotive crash relevant structures designated for absorbing crash energy during

deformation. In many cases of automotive applications, light weight structures are

additionally designed for stiffness, not allowing for significant bend [12].

Depending on the purpose of lightweight structures, two main construction principles

are employed overlapping each other at a certain level. One is the frame structure, only

carrying a given load, the other is a shell structure, which is used if the structure has to

seal against pressurised water, fuel, or air. In general, frameworks mainly involve the

use of beams like tubes or profiles, shell structures deal with sheet metal blanks. The

lightweight structures usually combine both concepts [13], and in most cases, straight

semi-finished products are joined to produce complex structures. Since steel is

relatively cheaper when compared to other metals with similar specific properties, at

present most light weight structures are made of steel.

1.4.1.1 Frame structures

Compared with shell structures, simple geometries at a small or medium scale are

used in frame structures and most of the frame structures are made from tubes with

round or rectangular cross-sections, e.g. axle tubes, bicycles frames, tubular frame

structure of motorcycles (Figure 1.2), etc. Unfortunately, they may show variations in

wall thickness of up to 20% [12]. Nevertheless, the extrusions offer excellent cross

section design possibilities to include additional functions [14], as a result, single hollow

extrusions are quite popular in small volume products as prototypes or niche cars in

automotive applications. At the same time, there is an increasing trend of using

aluminium extrusions to produce space frame body-in-white structures for specific

mechanical properties, such as the Ferrari F360 Modena (Figure 1.3), which has 44%

increase in tensional stiffness, and 42% increase in bending stiffness [15, 16].

Various materials are suitable for the tube extrusion of frame structures, especially

aluminium and magnesium alloys. In contrast, the normal steels used in the automotive

industry may not be extruded into hollow profiles with walls thin enough to meet car

body requirements. Hence space frames made of tailor steel blanks are considered

[17]. Such as laser welded steel tubes, either made by a continuous process, following

the conventional rolling of a sheet strip from a coil, welding and cutting, or

discontinuous second variant which uses sheet metal blanks bent into an open tube in

three steps, followed by welding. Actually, both of the process can be extended to

produce tailored blanks of different materials including stainless and high strength steel,

aluminium, or titanium [18].

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Figure 1.2 Tubular frame structure of BMW C1.

Figure 1.3 Profile based frame structure of the Ferrari F360 Modena.

In order to meet the increasing needs of frame structures, many forming techniques

related to frame structures have been improved. For instance, when bending complex

3D tubes and profiles, a new approach using a fixed tool and a moveable die

positioned in six axes which can be variably adjusted was developed, thus a workpiece

can be bent in a variable 3D shape [19]. Another kinematic approach in flexible 2D and

3D bending is using a polyurethane matrix [20]. In addition, based on the conventional

stretch bending, by applying adaptive tension force, the shape accuracy of the bent

profile can be greatly improved [21, 22]. Hydroforming of sheet metal pairs is also used

to produce sheet metal members for frame structures, which are conventionally deep

drawn using rigid punches and dies. Amongst all these, an innovative extrusion

process variant, named curved profile extrusion (CPE) was recently introduced which

can produce curved profiles directly at the press with greatly improved mechanical

properties compared with conventional stretch bent parts [23]. The CPE is based on

the design of an additional movable guiding tool, which is numerically controlled. An

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extra force from the guiding tool will be applied to the profile during the extrusion. As a

result, specific velocity profiles of the material flow on the inner and outer sides are

generated, due to the tensile and pressure stresses applied. Consequently, a curly

profile exits the die in a round shape, as shown in Figure 1.4 (left). This process also

allows for curved magnesium profiles, providing even higher potential for lightweight

applications.

Figure 1.4 Curved profile extrusion (rounding during extrusion) of a variably curved

bumper (left) compared with the stretch bent bumper (right).

1.4.1.2 Shell structures

Shell structures of automotive car body applications are usually for large-lot

production, forming thin walled hollow components with a surface quality suitable for

outer skin panels. Since the material price accounts for approximately 50% of the total

vehicle cost at large-lot production [24], steel is more commonly used than aluminium

and magnesium alloys. For crash relevant structures, ultra high strength steel is

preferred to improve crashworthiness, such as B-pillar of passenger cars in Figure 1.5.

Figure 1.5 High strength steel is employed in B-pillar of VW Passat.

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However, the employment of shell structures faces several challenges with the

development of automotive industry. From one hand, in order to cut the cost in

lightweight structures, larger sheet metal parts are favoured for shell structures, to save

the joining operations and shorten the processing chain. On the other hand, as new

and more complex materials are used, specific problems arise in deep drawing and

related processes when producing shell structures, such as the extreme sensitivity to

surface defects of aluminium due to its high adhesion tendency and the deposition of

workpiece swarf, the significant heavy load required to form high strength steels, and

the very low corrosion resistance and surface quality of magnesium sheet.

As consequence, corresponding advanced forming techniques are being developed

[13].

Firstly, combined with the use of tailored blanks, large single parts consisting of

different wall thicknesses, which must be assembled by different parts of various

thicknesses in the past, are achievable nowadays, e.g. side panel made of tailored

blanks as shown in Figure 1.6 [25].

Figure 1.6 A side panel made of tailored blanks.

Secondly, in the processing of aluminium sheet, adapted drawing depth, larger radii,

a homogeneous feed, and pressing the cutting punch into the workpiece surface are

considered. Additionally, applications of modern tool coatings, tool adjustment, and

lubricants have been implemented [26].

Thirdly, to avoid the disadvantages of magnesium sheet, a sandwich panel

comprising two thin layers either of steel or aluminium with a plastic layer in between

has been introduced. It has a density and shell properties comparable to magnesium,

but without the corrosion and surface quality disadvantages [17, 27]. Deep drawing of

sandwich panels can also be used for components requiring high stiffness, for example

aluminium foam sandwich for roof structures.

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Additionally, when forming high strength steels, the forming operation at elevated

temperatures is of increasing interest because it can significantly increase the

formability of a metal. Alternatively, it can be formed by using working media [28]. Such

as the popular hydro mechanical deep drawing (HMD) and high-pressure sheet metal

hydro forming (HBU) [29, 30] techniques. Use of the HMD and HBU processes allow

higher limiting drawing ratio to be achieved, and a better shape accuracy may be

attained compared with conventional deep drawing [31].

1.4.2 Light weight materials

Just as the world‟s auto industry is doubling efforts to reinvent itself, so too are its

primary materials suppliers, such as steel plants, aluminium and magnesium smelters,

and even plastics and composites producers, are reinventing themselves to meet the

daunting challenges faced by automakers worldwide. In order to meet the increasingly

severe requirements for weight reduction, aiming at the limitation of the fuel

consumption, an expansion in the range of application of light weight materials is vital.

Materials with high strength stiffness to weight ratio, good formability, good corrosion

resistance, and recycling potential are the ideal candidates to replace conventional

heavier or less strong materials in the car. As a consequence, the employment of

aluminium alloys, magnesium alloys, tailored blanks, high strength steels (HSS) and

advanced high strength steels (AHSS) have sharply increased in recent years.

In the current section light materials of AHSS, aluminium and magnesium are

reviewed.

1.4.2.1 Advanced high strength steels (AHSS)

“The global steel industry is continually reinventing its products to develop materials

that are lighter, safer, greener and cost effective,” said Edward Opbroek, Director,

WorldAutoSteel. “Steel, and particularly, Advanced High-Strength Steel, is the only

material that delivers improved environmental, safety and vehicle performance at little

or no cost penalty to manufacture.”[32]

Although various new materials have been developed and employed in the

automotive industry, steel has been the dominant one used in manufacturing

automotives since the 1920s, and always plays the most important role up till now.

It is proved that developing new high strength-mass ratio materials and improving

forming technique is most efficient to reduce car body weight. High strength steels and

advanced high strength steels were consequently designed to respond to the weight

reduction and increased safety requirement demanded over the automotive industry.

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The high strength steels are usually hardened by solid solution, precipitation or grain

refining. Contrastively, the advanced high strength steels are hardened by phase

transformation, which mainly have martensite, bainite, or small volume fraction of

retained austenite in the final phase. Figure 1.7 schematically compared AHSS to low

strength steels and traditional HSS [33], where IF is interstitial-free steel, MILD mild

steel, IS isotropic steel, IF-HS interstitial-free-high-strength steel, BH bake hardenable

steel, CMn Carbon-manganese steel, HSLA high strength low alloy steel, DP dual

phase steel, CP complex phase steel, TRIP transformation-induced plasticity steel, and

MS martensitic steel.

Figure 1.7 Schematic comparison among AHSS steels (TRIP, DP, CP, MS), low

strength steels (IF MILD, IS) and traditional HSS steels (IF-HS, BH, CMn, HSLA).

1.4.2.1.1 Dual phase (DP) steel

Dual phase steels consist of a ferritic matrix containing a hard second phase, e.g.

martensite or bainite, in the form of islands. The strength of DP is generally increased

with increasing fraction of second phase. Safety structural components for passenger

cars used to be made of dual phase steel consisting of 70% ferrite and 30% bainite

with a yield strength of 600MPa [34]. It is characterized as having excellent ductility and

high working hardening ratio due to the matrix microstructure, and has been widely

used in auto components that require high strength and formability, such as wheels,

bumpers and some reinforcement parts. The basic chemical composition of dual phase

steel has carbon and manganese. Extra chromium, and molybdenum, vanadium, and

nickel are occasionally added to enhance the hardenability.

1.4.2.1.2 Transformation-induced plasticity (TRIP) steel

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TRIP steel has a mixed microstructure of ferrite, bainite, martensite, and 5%~15%

retained austenite due to the silicon content. During deformation, the dispersion of hard

second phase in soft ferrite results in a high work hardening rate which is similar as to

that observed in the DP steels. Furthermore, the retained austenite progressively

transforms into martensite with increasing strain, therefore higher work hardening rates

are achieved at higher strain levels. It provides dramatic stretch formability when

compared with conventional HSS and a weight advantage over DP in the most severe

stretch forming applications.

TRIP steels can be hot rolled, cold rolled or hot dip galvanized with strength

primarily ranging from 600MPa~800MPa. It is suitable for stretch forming of complex

AHSS parts because of its high elongation and it is excellent in producing components

functioning as crash energy absorption due to its high work hardening rate. However,

the poor surface quality of TRIP steel sheet is difficult to improve.

1.4.2.1.3 Multiphase steel /Complex phase (CP) steel

Multiphase steel has a similar microstructure to the TRIP steel, but may additionally

contain pearlite, but have no or a very small volume fraction of retained austenite. The

multiphase steel has a strength mainly of 800MPa~1000MPa. Due to its high energy

absorption ability as well as high residual deformation capacity, it has been greatly

used for anti-crash rods, bumpers and B pillars.

1.4.2.1.4 Martensitic (MS) steel

Martensitic steel is a mainly lath martensite transformed from austenite by rapid

quenching. Small amounts of ferrite and/or bainite might exist. It thus has the highest

strength, ranging from 900MPa to 1200MPa, and can even reach an ultimate tensile

strength of 1700 MPa. MS steels are often subjected to post-quenching tempering to

improve ductility, and provide adequate formability. Compared with other types of

steels mentioned above, the very high strength of MS makes it more suitable for safety

critical components where the highest strength and stiffness are required in the

automotive industry.

There are other types of AHSS steels which may be less well known, but are

designed to meet specific processing requirements, such as ferritic-Bainitic (FB) steel

with improved stretchability of sheared edges, twinning-induced plasticity (TWIP) steel

combines extremely high stretchability and extremely high strength [35]. With the

various advantages reviewed above, although originally targeted only for chassis,

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suspension and body-in-white components, AHSS are now being applied to doors and

other body panels.

1.4.2.2 Aluminium and magnesium

To meet the need for weight reduction, aluminium and even magnesium, which are

more expensive than normal steel, are being considered for car body structures.

Aluminium alloy is the other type of most commonly used material in automotive

industry, for both shell and frame structures, apart from various grades of steels. The

main purpose of using aluminium alloys is for weight reduction. Such as the well known

extruded space frame developed for Audi A8 which reduced the body weight by 40%

(compared with the usage of steel), and the later model Audi AL2 has a modified space

frame with an all aluminium extrusions structure, with even fewer aluminium cast joints

(Figure 1.8) [36].

Figure 1.8 Audi AL2 with an all aluminium body structure.

The primary advantage of aluminium is the low density but equal mechanical

properties to some grades of steels, especially with the development of high strength

aluminium. It has been reported that up to 50% weight saving for the body-in-white

(BIW) applications can be achieved by the substitution of steel by aluminium, which

may result in a 20-30% total weight reduction [36]. Further more, aluminium may have

much better bare metal corrosion resistance (e.g. the series of 5xxx and 6xxx) than

steel, which leads to good paint durability without additional coatings. For instance, for

the skin sheet material, the bake hardening 6xxx alloys are the perfect choice for the

good combination of formability, strength after the paint-bake, and a high surface

quality after pressing and paint; for the structural sheet materials, 5xxx alloys are

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favoured because of the deep drawing behaviour and ability for impact energy

absorption. Also, both 5xxx alloys and 6xxx alloys have outstanding bare metal

corrosion resistance.

As a consequence, aluminium usage in automotive applications has grown sharply

in recent years, and is the greatest long-term threat to sheet steel. In 1996, a total of

110 Kg of aluminium is used per vehicle, this is expected to rise to 340 Kg by 2015 [36].

The applications of aluminium alloys in automobiles have covered a range of products

from chassis, and body structure to air conditioning systems. In addition, aluminium

alloys are the main materials that can be used in superplastic forming (SPF) for the

aerospace applications, trains or niche vehicles, targeting very complex shaped

structures. However, SPF is not suitable for mass production due to the long cycle

production times.

Magnesium offers a considerably higher specific strength compared to regular steel

and aluminium alloys. The application of magnesium may save the body weight up to

50% compared with steel and 20% with aluminium, for components without particular

strength requirements like front hoods, trunk lids, and doors. For example, in a

Volkswagen one-litre car model, 36 Kg of magnesium of thin-walled structures are

employed which saves 13 Kg of weight compared with the aluminium space-frame [37].

The main disadvantages of magnesium lie in the low corrosion resistance and

surface quality after deformation, as well as high price [38]. Great effort has been put

into the investigation of its potential application. Due to the light weight and high

strength properties, magnesium is a very promising and alternative material for

automobiles.

1.5 Challenges and objectives of the research

The strict environmental and safety requirements push car manufacturers to

continuously search for new solutions, in the direction of novel manufacturing

techniques and newly developed materials. Aluminium and magnesium alloys may

have some preferable properties compared with steel, however, are not competitive

from the aspects of cost and strength. Aluminium and magnesium alloys are generally

more expensive then steels, and less strong. The main challenge facing steel is how to

increase the strength-mass ratio, so as to reduce the car body weight and keep or

increase the structures strength.

AHSS is relatively new to the material world, but its application in automotive

structures has increased since the steel industry‟s UltraLight Steel Auto Body (ULSAB),

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UltraLight Steel Auto Closures (ULSAC) and ULSAB-AVC (Advanced Vehicle

Concepts) projects [32]. These programmes have successfully demonstrated weight

savings and performance improvements compared with similar applications using

conventional steels. Nevertheless, increased high forming forces at room temperature

due to the high strength of materials, additional cost on the post heat treatment, and

springback or the thermal distortion at high forming temperatures limited the further

application of AHSS. An advanced forming technique which can bring together the hot

forming, shape correction, and heat treatment on a single forming operation would

therefore be highly valued.

A forming process termed as hot stamping and cold die quenching has been

recently developed specifically for producing safety components for passenger cars,

and targeting low forming forces, enhanced strength after deformation, high efficiency

and accuracy. It enables the steel sheet to be formed in the austenite state with highest

formability, and quenched in cold dies to achieve a main phase of martensite for the

highest strength. Extra heat treatment is thereby saved and springback is minimized as

the formed part is firmly held in the closed die during the cooling process. The

combination of high formability, high strength and high shape accuracy is thus

achievable in a single operation.

Significant advantages of the hot stamping and cold die quenching process have led

to a wide utilization of the hot stamped ultra high strength steels in the automotive

industry for the manufacturing of safety critical component, such as bumper beams, A

and B pillars, side door impact beams and sills. The applications assist automotive

manufacturers in producing new, lighter safety critical panel components which achieve

higher strength body structures with lower weight, and result in fewer CO2 emissions

compared to conventional products. Just as importantly, the hot stamped products

allow for improved safety performance, while meeting manufacturing cost limitations.

The cost of hot stamped boron steel is much higher than convention steels employed in

automotive industry; however, they are significant lighter and safer, and in the long

term, it saves fuel and easily meets the exhaust emission legislation. For example, the

2008 Ford Fiesta, makes extensive use of ultra high-strength steels in its body

structure. Ford states that, “A remarkable amount of specialist steels, including boron

steel and dual-phase steel, is the secret to the Fiesta‟s quantum leap in structural

stiffness for its light weight.” This new vehicle promises to deliver CO2 emissions of less

than 100 g per km3 [39]. Similarly, the 2008 Mazda 2, whose body consists of more

than 40% high-strength steel, has a curb weight of 950 kg, 100 kg less than its

predecessor. It offers a 15% increase in fuel efficiency over its predecessor, with a

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corresponding reduction in CO2 emissions [40]. The 2008 Mercedes C-Class uses 70%

high-strength-steel alloys in its body, which results in a 9 tonne reduction in CO2

emissions per vehicle over its lifetime [41].

As a consequence of the new applications of hot stamped boron steels, a better

understanding of the steel behaviour, such as phase transformation, viscoplastic

properties and damage mechanisms, at each stage of the forming process is urgently

required. The boron steel, again, drew great attention among researchers and steel

companies due to its high quenchability, and has been developed to form safety

components using the hot stamping and cold die quenching process. Hot stamped

boron steel is rapidly becoming the predominant material in typical body structures,

reinforcing the need for a deeper understanding and relevant modelling techniques.

The research work described in this thesis deals with the microstructure evolution

and mechanical properties of high strength quenchable boron steel due to the thermal

and mechanical events occurring during the industrial hot stamping process. Both

experimental studies and modelling work were carried out.

1.6 Structure of thesis

This thesis has been divided into eight chapters. The first chapter is a short

introduction to the development of the modern automotive industry. The second one

elaborates the advanced technologies in forming high strength auto panels, in which a

newly developed forming process, known as hot stamping and cold die quenching, is

introduced and discussed. In chapter 3, a literature review on phase transformation

mechanisms and its modelling techniques are presented. In chapter 4, an experimental

study on austenitization of boron steel is described, and a set of unified constitutive

equations for modelling austenitization is presented, by which the incubation period

and volume fraction of growing austenite is rationalized, and effects of heating rate and

temperature are accounted for. Thermal mechanical property of the steel has been

investigated using a Gleeble thermal simulator. Formability tests using boron steel

sheets were conducted. The experiment apparatus, specimen designs, the stress

strain curves of hot tension tests and results of formability tests are introduced in

Chapter 5. A set of unified visco–plastic damage constitutive equations were developed

and determined and the modelling theories and techniques are presented in Chapter 6.

In Chapter 7, the unified constitutive equations describing bainite transformation during

quenching are presented. The developed model has been implemented into

commercial FE software, ABAQUS. The comparison of FE simulation of formability test

Page 37: modelling of phase transformation in hot stamping of boron steel

37

with the experimental results is described. The final conclusions are presented in the

last chapter, Chapter 8.

Page 38: modelling of phase transformation in hot stamping of boron steel

38

Chapter 2

Advanced technologies in forming

high strength panels for automobiles

2.1 Introduction

Many sheet metal forming techniques, such as bending, deep drawing, etc, have

been developed for forming a variety of sheet-metal components. In this chapter, sheet

metal forming techniques related to the production of parts made of high strength steel

(HSS) for automotive panel components are reviewed and improvements of traditional

sheet forming techniques for forming advanced high-strength steels (AHSS) are

described.

A new forming process, hot stamping and cold die quenching process (HSCDQ),

has been developed in order to meet the requirement of high strength/weight ratio in

producing high strength auto panels. The operational procedures, main process

parameters for the industrial practice, and advantages of this new process are listed

and application to the manufacture of parts in boron steel is reviewed. Challenge of this

process is analysed.

2.2 Traditional forming techniques

Forming of advanced high-strength steels is not a radical change from forming the

conventional high-strength steels (HSS). The principal difference between conventional

HSS and AHSS is their microstructure. Conventional HSS are single phase ferritic

steels, and AHSS are primarily multi-phase steels, which contain ferrite, martensite,

bainite, and/or retained austenite in quantities sufficient to produce unique mechanical

properties [32]. The acquisition of knowledge and experience needed for forming

higher strength steels in general has increased gradually over the years as increasing

strength became available in the high-strength low-alloy (HSLA) steels grades [35, 42-

49]. Although the demands for improved crash performance, reduced mass and cost

has spawned a new group of AHSS, the improved capabilities of AHSS did not bring

many new forming problems but accentuated problems that already existing with the

application of any HSS [35]. These concerns include higher loads on presses and tools,

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39

greater energy requirements, lower ductility, and increased need for springback

compensation and control [42, 45, 50]. Therefore, advanced technologies are needed

in the forming of AHSS.

In the processing of steel sheets, the use of HSS leads to significant challenges in

deep drawing, e.g. higher tool stresses, the significant development of wrinkles or

cracks, limiting the design possibilities [35]. The blank holder has to apply higher forces

in order to prevent wrinkling and this results in increased tool wear and necessitates

the use of premium tool material, tool coating or even ceramic inserts. The high yield

strength steels also cause a greater springback. Therefore, in addition to conventional

deep drawing, advanced techniques have been developed to assist the deep drawing

of HSS or AHSS. Using flexible working media is one approach to enable the

manufacture of complex sheet metal components [28, 51]. In pneumatic-mechanical

deep drawing, pneumatic preforming is employed to induce additional work hardening

in the sheet metal by stretching of the work piece centre, which brings a higher

buckling resistance and a more constant strain distribution [52]. In hydro mechanical

deep drawing (HMD), the die is replaced by a fluid [53]. A higher limit drawing ratio can

be achieved due to the greater surface pressure in a larger contact area between

punch and work piece, which increases the friction between the sheet metal wall and

the punch and thus the tear factor. The material fractures at a much higher stress value

than in the conventional case. An example of hydroforming of high strength steels is

the manufacturing of a fuel tank with a very complex geometry in [49, 54, 55]. Another

improved deep drawing technology was introduced by YOSHITAKE and YASUDA [56],

namely JFE Intelligent Multi-stage Forming with Press Motion Control. The trade name

is JIM-Form. In JIM-Form, the sliding operation of the press is controlled by a servo

press machine. The press motion is determined according to the friction behaviour

between the sheet and the die. This is a distinctive feature of this technology, on the

purpose of avoiding excess press load by optimizing the friction behaviour between the

sheet and the die in the press forming process. The newly developed method tends to

expand the forming limit by linkage with friction behaviour. It has been reported that,

when using a 780 MPa grade HSS, the maximum press load at a give blank holding

force was reduced by approximately 10%, and consequently it improved the forming

limit of the material [56]. As a consequence, the technology can be applied to materials

with a strength up to 980 MPa, such as AHSS and to hard-to-form products which

could only be press-formed with 780 MPa grade materials in the past [56]. The

drawback of this new technology is that the productivity is less, compared with

conventional deep-drawing.

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Roll formed HSS and AHSS have been widely used for the productions of side door

impact beams and B-pillars, as well as other chassis members and reinforcements.

Roll forming is quite suitable for producing complex shaped components from AHSS,

especially those with small total elongations such as Martensitic steels (MS). Roll

forming can be used to produce AHSS parts with small radii depending on the

thickness and the mechanical properties of the steel [32]. However, the forces on the

rollers and frames are quite high; as an approximation, the force is proportional to steel

strength but increases as the square of the thickness. Since the cold formability of HSS

and AHSS is comparatively poor, even the roll forming ability is limited. From this point

of view, laser-assisted bending and roll forming leads to the local reduction of flow

stress and thus improves roll formability [57]. Laser assisted bending aims at local

softening of the laser heated high-strength steel during bending, but avoids permanent

softening; thereby dramatically decreasing punch load while increasing attainable

bending angle, see (Figure 2.1) [57]. It is reported that the punch force required for

laser-assisted bending is around 30% smaller than the punch force necessary for the

regular bending process [57]. The amount of increased bending angle depends on the

material, the orientation of the punch to the rolling direction, the size and geometry of

the punch and laser parameters, such as laser power and the number of irradiations.

Figure 2.1 Comparison of maximum bending angle of MS 1200 with (right)/ without (left)

laser-assisted bending [57].

A load-adapted material distribution is a key to successful lightweight auto panels

[46] and tailored metal sheet has been developed for this purpose. A tailored sheet is a

metal piece in which different wall thicknesses (tailored blanks), different material or

alloys (tailored heat treated blanks), or even grades (hybrid blanks) are combined

within a single workpiece. The use of tailor-made semi-finished product enables the

cost efficient production of weight and load-optimised components, as well as

sophisticated products with complex geometries or improved properties [12, 46]. The

manufacture of crash absorbing structures for the automotive industry using tailor rolled

Page 41: modelling of phase transformation in hot stamping of boron steel

41

blanks (TRB) not only reduced weight by 13%, but also improved structural behaviour

significantly in static and dynamic load tests [46]. However, manufacture of such

complex semi-finished products calls for adapted forming strategies which require

increased process knowledge, the observation of different material behaviours, and the

development of designated adaptive forming processes and tools.

High frequency welded tubes have been widely employed in auto panels, such as

seat structures, cross members, side door impact beams, bumpers, engine subframes,

trailing arms, and twist beams. Operations for forming tubes into automotive

applications include: flaring, flattening, expansion, reduction, die forming, bending,

hydroforming, or combined operations. Post trimming or piercing might be required for

tube forming. Since AHSS tubes have higher tensile strength, trim tools that can

withstand larger loads are necessary. Laser trimming is commonly used for

hydroformed parts. AHSS tubes also provide high energy absorption and cost efficient

manufacturing. Figure 2.2 gives an example of the forming of an Engine Cradle, using

AHSS tubes [32], which has greatly saved the total mass, and increased strength

compared with a conventional Engine Cradle, made of HSLA steel. However, AHSS

tubes have limited elongation and forming limits are often exceeded.

Figure 2.2 Hydroformed Engine Cradle made from welded DP 280/600 tube. Draw

bending, Bending Angle>90°[32].

There are also other types of forming techniques available for HSS or AHSS. Such

as laser welded tailored tubes, which are commonly used to produce components with

complex variations in shape, thickness, and strength [58, 59]. Superplastic forming

(SPF) enables complex-shaped components to be formed in one operation [60]. But

SPF is too slow for mass production.

In general, conventional cold forming of high strength steel is limited to the

production of relatively simple geometries, due to the material‟s low formability. In

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42

addition, high flow stress, large springback, low efficiency and excessive tool wear

often arise. Alternatively, it is possible to form a high strength steel at elevated

temperatures, followed by fixing (to correct excessive thermal distortion) and heat

treatment (to harden the material). Such operations are costly and not suitable for large

scale production. However, the SPF process illustrates the feasibility of forming and

heat-treating together, so that a metal can be formed at low strength but high ductility,

and attains higher strength on finishing. This could be a very effective way of

increasing the strength-mass ratio, thus a green and environment friendly forming

process becomes quite promising.

2.3 Hot stamping and cold die quenching

Ideally, components are formed with materials having high ductility and low strength,

but hardened during processing. It is common knowledge that, in general, metals are

softened at elevated temperature. But in hot/warm forming processes, the original

microstructure of materials might be destroyed and subsequent restorative heat-

treatment for formed sheet-metal components could cause significant distortion [45].

Additionally operations, such as quenching and tempering, have to be carried out to

harden the material or improve the material properties, which are costly for mass

production. In order to solve these problems faced in hot forming, a new forming

process termed as hot stamping and cold die quenching process (HSCDQ) has been

developed. For this process to be effective steel of high hardenability is required, and

this has become available in the form of recently developed low carbon boron steels.

2.3.1 Coated boron steel

The advantages of the hot stamping and cold die quenching are directly linked to the

high hardenability and uniform microstructure of the steel sheet blank. Boron steel,

which is known for its splendid hardenability is employed in the process. The

considerable interest shown in boron steels is based on the fact that a minute amount

of boron, around 0.002%-0.005%, may increase twofold the effect on hardenability of

the other alloying elements. Figure 2.3 is the Continuous cooling diagram (CCT) of a

boron steel, USIBOR 1500 P®, which is the investigated steel in this thesis, developed

by ArcelorMittal particularly for the hot stamping and cold die quenching process. The

alloy composition is given in Table 2.1. It is shown in Figure 2.3 that, the critical

martensitic cooling rate is 27℃/s for boron steel. However, a cooling rate of 300℃/s is

required to ensure full transformation to martensite for some normal low carbon steels

[61], which is unachievable by cold die quenching.

Page 43: modelling of phase transformation in hot stamping of boron steel

43

Figure 2.3 CCT diagram of USIBOR 1500 P provided by ArcelorMittal.

Table 2.1 Chemical composition of USIBOR 1500 P (Maximum %).

C Mn Si Sr Ti B

0.25 1.40 0.35 0.30 0.05 0.005

Weight reductions for the body-in-white through the use of high-strength steel sheet

can only progress further with the realization of the need for high-strength steel sheets

combining excellent press formability and sufficient coated surface quality for exposed

automobile body panels [44]. Because of the high temperature and unavoidable

contact with air during transfer from the furnace into the stamping press, the surface of

the hot blank is subjected to oxidization, unless pre-coated. An aluminium-silicon

coating developed for the USIBOR 1500 P provides excellent protection from both

scale formation and decarburization. This metallic coating has a thickness between 23

μm and 32 μm, which can transform into an alloy, made of Fe-Al-Si and strongly

adherent on the steel sheet substrate, during the heat treatment. The pre-coated boron

steel is ready for painting after hot stamping, thus shortening the process sequence, as

the conventional shot blasting operation is saved [34, 50].

2.3.2 Forming process

Figure 2.4 schematically illustrates the whole process of HSCDQ, with the

corresponding temperature profile. It is shown in the figure that the sheet steel blank is

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44

first austenitized in a furnace. The furnace temperature is above 900℃. When the steel

is fully austenitized, the hot blank is quickly transferred to a cold die for simultaneous

forming and quenching. A cooling system is often used to guarantee a sufficient cooling

rate between the closed dies. The formed components are firmly held in the closed

dies until the blank temperature drops below 200℃, when phase transformation ceases.

The microstructure of the formed part comprises largely martensite.

Figure 2.4 Temperature profile and illustration of the HSCDQ process. Microstructures

at different stages are indicated.

Hot stamping of sheet metals enables metals to be formed at low strength and high

ductility. When the metal, e.g. steel with high hardenability, is formed and held in a

cold-die set, a good cooling rate is achieved, which enables a transformation from

austenite to pearlite, bainite or even martensite, with very high strength. In addition, the

springback and distortion due to rapid cooling and phase transformation can be

eliminated. Geometry accuracy of the products can be significantly improved. In order

to maximise the potential of the process, full transformation to martensite is required,

so that the strongest components can be formed.

Ferrite and pearlite

Page 45: modelling of phase transformation in hot stamping of boron steel

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Important process parameters for the industrial processing of boron steel are listed

below:

a) Heat treatment condition

The austenitization condition in the heating furnace is performed with respect to the

metallurgical transformation and the intermetallic alloying reaction between the iron

substrate and the Al-Si coating which is unique for the USIBOR coated steel. The

maximum heating rate is 12℃/s so as to allow the alloying reaction and preserve the

layer integrity [50]. Austenitization temperature ranges from 880℃ to 940℃ for a

soaking time from 4 to 10mins, so that a homogeneous austenite phase is achieved,

and abnormal or excessive grain growth is avoided.

b) Transfer from heating furnace to press

The transfer operation is usually fast in order to minimize the heat loss and keep the

elevated temperature of the heated blank, which avoids the formability reduction and

the occurrence of local phase transformation. In practice, the transfer time is shorter

than 7s.

c) Forming speed

A fast tool closing speed, e.g. 100 mm/s, is necessary to reduce heat exchange

between blanks and dies during deformation, so that the steel formability is high and its

flow stress is low.

d) Quenching

The quenching speed in the cold dies has to be sufficiently high (minimum 27℃/s for

boron steel) so that single phase martensite can be obtained. So for mass production

the hot stamping dies are therefore water cooled. Water quenching and post tempering

might be required to improve the microstructure in the final formed parts. A formed

component is removed from cold dies when its temperature is below 200℃, hence the

springback can be minimized and no phase transformation occurs after removal.

e) Trimming

Due to the high strength, laser trimming is employed. Alternatively, die-trimming with

low clearance can be used to shorten the process sequence.

2.3.3 Applications and market trends

The potential application of hot stamped boron steel is even wider than conventional

HSS and AHSS, especially when combined with tailored blanks, and almost covers the

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46

whole range of reinforcement structures in vehicles, as shown in Figure 2.5 [42, 43].

This is because the use of Tailor Welded Blanks (TWB) can significantly improve the

crash worthiness, as the material thickness, strength and elongation are locally

adjusted [43, 62].

Figure 2.5 Potential applications of hot stamped laser welded blanks combining

USIBOR 1500 P and DUCTIBOR 500 P [43].

Figure 2.6 is a comparison of mechanical properties (yield strength and ultimate

tensile strength) of different types of HSS and AHSS with HSCDQ boron steel.

Compared with the conventional HSS/AHSS, hot stamped of boron steel has:

a) A much higher YS and UTS (1500 MPa) can be achieved. This can significantly

contribute to the foremost goals of automotive industry: reduced weight with increased

crashworthiness.

b) Complex geometry structures can be formed in a single punch with low

stamping force due to the high formability and low flow stress of the heated steel.

c) Springback is minimized, which dramatically improves forming accuracy.

d) High efficiency is achieved compared with the conventional multi-stage forming

process, as the forming and quenching are simultaneously accomplished in cold dies,

and additional operations are saved, such as shot blasting.

Body Side Inner

Door Frame

Roof Reinforcement

B-Pillar

Side Member

Bumper Beam

Page 47: modelling of phase transformation in hot stamping of boron steel

47

Figure 2.6 Comparison of yield strength and ultimate tensile strength of different types

of hot stamped steel, HSS steels and other types of AHSS steels.

Thanks to these advantages, the market for the hot stamped structures employed in

automotive industry has expanded fast. In Europe, the estimated total consumption of

flat boron steels for HSCDQ was around 60000 to 80000 t/year in 2004, and has more

than doubled, to 300000 t/year, in 2009 [34, 43, 50]. The targeted components have

covered most of the crash relevant parts of car structures, e.g. bumpers, side impact

reinforcements, A- and B-pillars, tunnels and front/rear members. Table 2.2

summarizes the growth of the hot stamping technology in Europe [43, 50].

Table 2.2 Growth of hot stamping technology in Europe [43].

Year of investigation 2003 2009

Hot stamping lines in Europe 15 42

Hot stampers in Europe (incl. OEM‟s) 3 10

Car models with hot stamped B-pillar 7 20

Hot stamped B-pillars in European car models produced in USIBOR 1500 P 0 11

Yie

ld S

tre

ng

th (

MP

a)

Ultimate Tensile Strength (MPa)

900

800

700

600

500

400

300

1400

1200

1100

1000

1300

300 500 700 900 1100 1300 1500 1700

Martensitic Steels

Multiphase Steels

Dualphase Steels

Trip Steels

USIBOR 1500 P before hot stamp

Hot stamped USIBOR 1500 P

Page 48: modelling of phase transformation in hot stamping of boron steel

48

2.4 Challenges of hot stamping and cold die quenching

process

There are quite a few uncertainties in HSCDQ as it is a new process and is affected

by various parameters. A good understanding of the involved phenomena is necessary

for a scientific understanding and an ability to accurately control the process.

Key problems requiring greater understanding of the process are: austenite

transformation, material formability, and martensite transformation.

Intensive study of the steel behaviour, such as phase transformation, formability and

damage mechanisms, at each stage during the forming process is needed. An

accurate mathematical transcription of the coupling between the thermal, mechanical

and metallurgical issues will be helpful to both the steel suppliers and automotive car

makers. Development of relevant modelling techniques for the new forming process is

important. Currently, no modelling technique is available to describe the entire process,

including austenitization, mechanical behaviour of the steel, and bainite transformation

with strain effect. Therefore, constitutive equations which describe the interactive

effects of viscoplastic deformation, microstructure evolution and phase transformation,

are most desirable as convenient pre- and post- processing tools, for process planning

and product quality control. However these equations are difficult to determine and

validate [63].

The research work described in this thesis deals with modelling the hot stamping

and cold die quenching process, and is targeted at the development of unified

constitutive equations describing coupled thermal and mechanical behaviour of the

steel, the phase transformations under certain heating, soaking and cooling conditions,

and the evolution of microstructure as a function of temperature, time and deformation.

Formability tests were employed to valid the proposed models. The investigated

material is the Al-Si pre-coated boron steel, commercially named as USIBOR1500P,

produced by ArcelorMittalTM.

Page 49: modelling of phase transformation in hot stamping of boron steel

49

Chapter 3

Phase transformation and modelling

methods

3.1 Introduction

The ability to predict and control on-line work-piece microstructure and mechanical

properties in metal processing has become a necessary production feature. Thus steel

producers put significant effort into the development of computer prediction and control

systems [64]. In this chapter, austenite and bainite transformation mechanisms and

models are analysed and discussed.

3.2 Phase transformation mechanisms

3.2.1 Austenitization

The mechanism of austenitization of steel is reviewed in this section. The author‟s

paper: Austenitization mechanisms and modelling methods for steels [65], contains a

more detailed discussion of the subject.

Typically, the austenitization process can be divided into three steps: nucleation,

growth and homogenization [65, 66]. Formation of austenite depends greatly on the

parent phase prior to heating. Grain growth and homogenization is a carbon diffusion

controlled process, which depends significantly on microstructure, chemical

composition, soaking temperature and soaking time [67-78].

3.2.1.1 Nucleation

From the relationship between the amount of formed austenite with austenitizing

time for eutectoid steel [69], it is shown that there is an incubation period for the

formation of the first austenite nucleus, after which more nuclei develop and grow at a

much higher rate. The incubation time is required in order to satisfy the thermodynamic

condition for nucleation [79]. The eutectoid reaction of ferrite + carbide → austenite

occurs when austenite nucleates. Therefore, austenite is expected to nucleate at

carbide-ferrite interfaces. As the driving force for heterogeneous grain nucleation is

determined by thermodynamics and geometry, the surface energy at the interfaces of

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50

parent and product phases has to be taken into account. When the temperature

increases to that at which austenitization occurs, the free energy change increases with

the increasing temperature, resulting in an increasing rate of austenite nucleation.

Therefore, the starting and completion of austenitization will be earlier at a higher

temperature than at a lower one for an isothermal case, as shown in Figure 3.1 where

, , represent austenite, cementite and ferrite regions, respectively; C% is average

carbon concentration; C

(%), carbon concentration in austenite at the / interface;

and C

(%), carbon concentration in austenite at the / interface.

Geometrically, the surface energy is a function of the lattice mismatch between crystals,

thus a high angle parent grain boundary would be more favourable for austenite

nucleation [68, 80, 81]. Figure 3.2 schematically illustrates the preferred nucleation

sites for austenite formation from the parent phase of ferrite, ferrite with spheroidized

cementite and lamellar pearlite. Experimental evidence of austenite nucleation at

interfaces of a pearlite matrix or a primary ferrite matrix can be found in [66].

Page 51: modelling of phase transformation in hot stamping of boron steel

51

C

completion of austenite formation

start of austenite formation

time

T(℃)T(℃)

0.80 6.67 C%

C

γ γ+θ

α+γ

α θ

α+θ

α+γ

γ α

a) Isothermal TTT diagram of the formation of austenite of eutectoid steel (C% = 0.8).

completion of austenite formation

start of austenite formation

time0 0.8 6.67

T(℃)T(℃)

C%

α

γ γ+θ

α+θ θ

α

α+γ

γ

α+γ

b) Isothermal TTT diagram of the formation of austenite of hypoeutectic steel (C < 0.8).

Figure 3.1 Schematic diagram of an isothermal TTT diagram of the formation of

austenite. a) Eutectoid steel (C% = 0.8); b) Hypoeutectic steel (C% < 0.8).

Austenite

Carbide

AusteniteFerrite

Ferrite

a) b) c)

Figure 3.2 The preferred nucleation sites for austenite formation from the parent phase

of a) pure ferrite, b) ferrite with spheroidized cementite and c) lamellar pearlite.

Page 52: modelling of phase transformation in hot stamping of boron steel

52

3.2.1.2 Grain growth of austenite

After the nucleation of austenite, further austenitization will be determined by grain

growth. For an Fe-C steel, grain growth of austenite is controlled by diffusion of carbon

at the / interface and the / interface, thus it is greatly affected by factors of

temperature, material composition and grain size of parent phases. The carbon

distribution is uneven in the growing austenite [71, 72, 75, 76, 78, 82]. The evolution of

carbon concentration and carbon distribution profile in each phase is schematically

shown in Figure 3.3. r and r are the positions of / interface and / interface,

respectively; r0 is the initial / interface position, which is equal to the initial radius of

the assumed spherical cementite particle.

3.2.1.3 Homogenization of austenite

Homogenization of austenite includes the processes of dissolution of retained

cementite and chemical homogenization. Behind the moving interface of austenite-

parent phases, austenite may contain undissolved carbide in the form of spheroidized

particles [81, 83, 84], termed retained cementite. Immediately after the parent phase

disappears and the steel is fully transformed into austenite, the chemical composition is

still not uniform. The development of the austenite grain size and carbon concentration

profile during the whole austenitization process is illustrated in Figure 3.3, for a low

carbon steel whose average carbon content is C0. r is the distance from the zero point.

Stage Ⅰ represents the initial state of parent phases composed of cementite and

ferrite; Stage Ⅱ represents the growth of austenite, Stage Ⅲ represents the

homogenization process and Stage Ⅳ is the final homogenized austenite. The

corresponding carbon distribution profile is shown in Figure 3.3 b). A significant carbon

concentration gradient exists in the austenite region. Curve Ⅲ, corresponds to the

austenitization stage of Ⅲ in Figure 3.3 a). To produce an acceptable homogenized

microstructure, the steel has to be reheated or held in the austenite temperature region

for a length of time sufficient for dissolving retained cementite and eliminating uneven

chemical composition in austenite. A uniform carbon distribution in austenite, which is

equal to the average carbon content in the virgin material, will be the final result (Curve

Ⅳ in Figure 3.3 b). This process is termed homogenization, and is a carbon diffusion

controlled process for the Fe-C steel. Compared with austenite nucleation and grain

growth, homogenization is usually far slower. 6.67% is the carbon content in cementite

( ).

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53

Figure 3.3 a) Illustration of austenite nucleation, grain growth and homogenization in a

two-phase steel; b) Carbon distribution profile in each phase at various stages

of austenitization.

3.2.2 Bainite transformation

Bainite was first discovered in the late 1920s by Davenport and Bain, at

temperatures above that at which martensite begins to form, but below those at which

fine pearlite is found, when studying the isothermal transformation of austenite. [85, 86].

Bainite is a two-phase mixture of ferrite and cementite which is distinguishable from

pearlite, and the bainite C curve may overlap with the pearlite curve in a CCT diagram

of steel, consequently both bainite and pearlite form competitively at around 500℃ in

plain carbon steels. The microstructure of bainite depends mainly on the temperature

at which it forms [87], and the bainite start (Bs) temperature is quite sensitive to the

steel composition. An empirical equation for the bainite start temperature is given by

Steven and Haynes as [88]:

( ) 830 270 90 37 70 83s C Mn Ni Cr MoB C

where i is the wt% of element i in solid solution in austenite.

α

α

a) Original

Complete

γ θ

θ

γ

γ

6.67

C%

0 distance r

a

b)

C

C

C

C

r r0r

0C

r0

rγθ rγα

Page 54: modelling of phase transformation in hot stamping of boron steel

54

In the temperature range 350℃-550℃ for plain steels, bainite consists of needles or

laths of ferrite with cementite precipitates between them. This is termed as upper

bainite. The ferrite nucleates on austenite grain boundaries with a Kurdjumov-Sachs

(KS) orientation relationship, which is specified as {110}bcc/{111}fcc,

<111>bcc//<101>fcc, and then grows into the austenite. As the laths thicken, the

carbon content of the austenite increases and finally reaches such a level that

cementite nucleates and grows. At sufficiently low temperatures the microstructure of

bainite changes from laths into plates and the carbide dispersion becomes much finer.

This type of bainite is named lower bainite. The diffusion of carbon is relatively slow in

lower bainite due to the low temperature range, and carbides (either cementite or ε-

carbides) precipitate in the ferrite with an orientation relationship, aligning at

approximately the same angle to the plane of the ferrite plate [89]. However, some

carbide can also be found between the ferrite plates. The transition between these two

forms is not sharply defined, and microstructures may contain either upper bainite or

lower bainite, or both forms. Figure 3.4 illustrates carbide precipitation from austenite

into upper and lower bainite [90]. Figure 3.5 a) [91] and b) [92] illustrate the

microstructures of upper and lower bainite, respectively.

Figure 3.4 Schematic illustration of the development of upper and lower bainite in

steels [90].

Page 55: modelling of phase transformation in hot stamping of boron steel

55

a) b)

Figure 3.5 Transmission electron micrographs illustrating a) upper [91] and b) lower

bainite [92].

Formation of bainite in steels requires redistribution of carbon and is thus diffusional,

but not necessarily diffusion controlled. Many measurements have shown that the

growth rate of bainite it too rapid to be attributed to carbon diffusion [79]. There have

been many arguments on whether bainite transformation is a diffusional process, or a

martensitic transformation [93-95]. As early as the 1950s, it was observed that bainite

and Widmanstatten ferrite in a hypereutectoid steel grow simultaneously and produce a

martensite-like upheaval on the free surface of specimens, but grain boundary ferrite

allotriomorphs do not. Bainite forms by shear as does martensite, but accompanied by

diffusion of carbon in the parent phase [96]. More experimental results have shown that

formation of bainitic ferrite at a free surface gives rise to surface relief, and it has been

well-demonstrated that the shape change has the characteristics of invariant plane

strain (IPS) [97]. It has thus been proved that transformation to bainitic ferrite is

displacive and that the lattice deformation within the transformed volume has a

significant component of shear. Accounting for the significant diffusion of carbon

involved, bainite transformation is then widely accepted as being diffusional-displacive

[97-100].

3.3 Modelling of phase transformations

When a hot stamping operation is in the planning stage, three aspects are of

particular interest: feasibility of the forming (accurate geometry without wrinkling or

tearing), microstructure after the forming and quenching, and process data, such as

cycle times, press force, and thermal load on the dies [34]. These can be predicted,

0.5μm 0.5μm

Cementite

Ferrite

Page 56: modelling of phase transformation in hot stamping of boron steel

56

using computer-based modelling and simulation techniques, as demonstrated below.

But first, the state of art of relevant modelling techniques is reviewed in this section.

3.3.1 Modelling of austenitization

3.3.1.1 Kinetic model

Tszeng and Shi has developed an algorithm to identify the overall kinetics of

austenitization based on fitting equations to dilatometry data [101]. The kinetic model

employed for the nucleation and growth of austenite is developed from the Avrami

equation (Equation 3.1):

211 exp

Af At

3.1

where f is the transformed volume fraction of austenite, t is time and 1A , 2A are

temperature dependent parameters. The transformed austenite fraction was thus

introduced as

1

0 01 exp exp exp1

NN VNQG t

f I gRT RT N

3.2

where G is the energy barrier for stable austenite nucleation, VQ is the activation

energy of austenite grain growth. R is the universal gas constant, 0I and 0g are

material constants and N is a factor that depends on the morphology of the new

phase.

The model describes the relationship of transformed austenite fraction with

increasing temperature and Equation 3.2 can be calibrated by fitting it to the

dilatometry curve and evolutionary curve of austenite fraction vs. temperature obtained

during the austenitization process. A TTT diagram has been generated by Tszeng and

Shi [101] using the above model. For certain types of steel, the equation can fit the

dilatometry data and austenite fraction curve very well if the steel is heated at a

constant heating rate [101]. However, the model does not describe the incubation

period. Since austenitization of steel is a carbon diffusion controlled process, both

temperature and time are key factors, heating rate and incubation period are important

in describing the rate of austenitization. Within a particular temperature rage, a faster

heating process is expected to result in less volume fraction of austenite compared with

that in a slower heating process because there is less time for carbon to diffuse. In

addition, if the incubation period is ignored, the start temperature of austenitization

Page 57: modelling of phase transformation in hot stamping of boron steel

57

process cannot be accurately predicted. Further more, the existence in the equation of

the parameter N which depends on the morphology of new phase, requires that

microstructure be considered. This brings an extra constraint in using the model. A

model that can combine both time and temperature dependency of austenitization is

favoured, for a more accurate description of the process.

3.3.1.2 Modelling of austenitization based on Fick’s second law

The majority of models for predicting the kinetics of austenite transformation are

based on nucleation, grain growth and homogenization. Some neglect the nucleation

step, as the nucleation of austenite in an Fe-C steel takes place just above the

eutectoid temperature, and the nucleation sites are very quickly saturated. The

subsequent austenitization process is then dominated by grain growth and

homogenization. As a consequence, many austenitization models concentrate on

austenite grain growth and homogenization [68, 70, 73, 77, 84, 101-105].

As austenitization is diffusion controlled, Fick‟s second law, in the form of partial

differential equations, is wildly employed in modelling of austenitization [71, 72, 74-76,

78, 106, 107]. Basic assumptions are usually introduced to simplify the modelling work,

for example, in Fe-C and Fe-C-alloy systems.

3.3.1.2.1 Austenitization between and phases

In a ferrite-cementite system, austenite is believed to nucleate

instantaneously at the carbide-ferrite interfaces, followed by diffusion controlled

austenite grain growth. This is illustrated in Figure 3.3. In order to simplify the model, it

is assumed that [70, 73, 74, 77]:

1) Local equilibrium is held at all ferrite-austenite and cementite-austenite phase

boundaries;

2) The diffusion of carbon in ferrite and cementite can be ignored;

3) The effect of capillarity can be neglected;

4) The diffusivity of carbon is concentration and temperature dependent.

5) The effect of capillarity on the dissolution rate of the carbides can be neglected.

Consequently, the growth of austenite, which is carbon diffusion controlled, can be

described by Fick‟s second law applied with correct initial and boundary conditions.

The carbon concentration Cr in austenite is a function of time t and distance from the

centre of the cementite particle r [66, 71]:

Page 58: modelling of phase transformation in hot stamping of boron steel

58

n

n

D CCr

t r r r 3.3

where D is the carbon diffusion coefficient in austenite which is a function of carbon

concentration of Cr and temperature T. The evaluation of carbon diffusion coefficient

has been discussed elsewhere [71, 72, 75, 76, 78, 105-108]. Usually, it is assumed to

be concentration dependent for an isothermal austenitization process and temperature

and concentration dependent for a thermal austenitization process. n is an integer

related to the geometry of an austenitization model; n = 0 for planar geometry, n = 1 for

cylindrical geometry and n = 2 for spherical geometry.

At the cementite/austenite interface:

r r

dr CC C D

dt r

3.4

At the austenite/ferrite interface:

r r

dr CC C D

dt r

3.5

where r and r are the instantaneous positions of the / and / interfaces

respectively (Figure 3.3). The superscripts “+” and “-” indicate the directions of / C r

to the austenite region at r and r . C

and C

are carbon content in austenite at

the / and / interfaces, which can be determined by the numerical minimization

of Gibbs energy [73]. The initial boundary conditions (t=0) are given as

00 0t tr r r

; C C for 00 r r and C C for 0r r where r0 is the initial

radius of the cementite particle. Local equilibrium at the / and / interfaces are

r rC C

and

r rC C

.

The carbon concentration decreases in the growing austenite region, which is from

C

to C

, and follows the leverage rules in the Fe-C diagram as shown in Figure 3.1

a). The value of the carbon concentration drop at the / and / interfaces

depends on the austenitization temperature. Numerical and analytical solutions of the

above equations for planar and spherical geometry can be found in [70, 73, 74]. It

should be noted that, there must be a carbon concentration gradient in the ferrite

region as well due to the carbon diffusion at / interfaces, as shown in Figure 3.3 b).

This carbon concentration gradient is usually neglected due to the low solubility and

Page 59: modelling of phase transformation in hot stamping of boron steel

59

diffusivity of carbon in ferrite [70, 73, 74, 77]. The assumption is acceptable for steels

with relatively high carbon content, but may not be accurate enough for a low carbon

micro-alloyed steel, which contains mainly ferrite. The carbon diffusion in ferrite plays a

much more important role in this case. However, little effort has been made to

investigate the carbon concentration gradient in the ferrite phase during austenitization

of steels and no material model is available for this purpose.

It may happen that the cementite in the parent material has not been fully dissolved

even though the ferrite has been fully austenitized. This can be seen, in Figure 3.3, that

when r reaches the cell boundary a, r has not reach the symmetry axis (r=0) at the

same time. In such cases, extra time is required for the dissolution of the retained

cementite and austenite to homogenize. Compared with the time required for

nucleation and grain growth of austenite, the homogenization process takes much

longer. Fick‟s second law can be applied to model austenite homogenization which is

carbon diffusion controlled. In Karlsson and Larsson‟s model [71, 72], homogenization

of austenite can be described by the same set of equations as that used to model

austenite grain growth. The only difference is the application of different boundary and

interface conditions.

3.3.1.2.2 Austenitization between / or θ / phases.

For a general case, it is assumed in the model that the ranges of existence of the

two phases, and , or θ and phases, are limited and separate. The diffusion of

carbon in ferrite and cementite is ignored. The concentration at the interface of both

phases is considered constant (the transformation at the interface is assumed infinitely

rapid) and no mass transport across the cell boundary occurs, and molar volume is the

same for all , θ and phases. If use to represent a single phase or phase,

then in phase:

1 n

n

C Cr D C

t r r r

0<r<y(t) 3.6

where y(t) is a function of time, calculating the interface coordinate: r=y(t). The

interface mass balance is r yr y

C CdyC C D D

dt r r

; The

boundary conditions are

0,

0

r a

C

r

, ,C y t t C

and ,C y t t C

; The

initial condition is 00 0

,0r y r

C r C and

00,0

y r r a

C r C , where a is the cell

Page 60: modelling of phase transformation in hot stamping of boron steel

60

boundary, r0 is the initial interface position of the and phases. A numerical

solution of Equation 3.6 is given in [71].

After the interface reaches the cell boundary (x=a), the remaining diffusion is limited

to the phase, until a homogeneous phase is obtained. The initial carbon

concentration profiles and interface concentrations are taken from the Fe-C phase

diagram; mean composition, cell size, diffusion coefficients and node spacing are pre-

defined before the modelling work.

The above model can also be extended to cases when pearlite is included in parent

phases, for example lamellar structured pearlite and ferrite (Figure 3.6). In a pearlite

grain, there are many interfaces. The diffusion of carbon in the layered pearlite is

at a nano-scale. Austenite can nucleate at the various interfaces simultaneously,

thus all pearlite grains are austenitized shortly after the temperature reaches the

austenitization temperature [109]. The forming and growing rates of austenite in a

lamellar pearlite structure is much faster than that in ferrite. As a consequence,

austenitization mainly occurs in the phases of austenite which are formed from pearlite

with a carbon concentration of 0.8% and ferrite exists in parent phases (Figure 3.6),

neglecting austenitization of pearlite. The problem is thus converted into austenite grain

growth and homogenization in a system, and can be calculated by the model

introduced in this section.

Page 61: modelling of phase transformation in hot stamping of boron steel

61

Figure 3.6 Illustration of austenitization for a pearlite and ferrite parent material. a)

Parent phase; b) Grain growth of austenite; c) Carbon concentration profile when t=0.

Figure 3.7 shows the calculated carbon concentration profiles in growing austenite

at different time stages for the case shown in Figure 3.6, employing the 2D model by

Karlsson [71], as given by Equation 3.6. In the initial condition, austenite grain size is

given as 4μm, which equals the original pearlite grain size in the parent material. The

ferrite grain boundary to the centre of symmetry is 7.06μm. The value of carbon

diffusion coefficient is measured from the experimental curve of carbon concentration

dependence on the diffusion coefficient when the austenitization temperature is 758℃

a)

Ferrite Pearlite

t=0t>0

b)

AusteniteFerrite

y(0) y(t)

α

α γ

γ

θ

θ

0.8C%

0 distance ry(0)=r0 a

0.022

c)

γ

α

Page 62: modelling of phase transformation in hot stamping of boron steel

62

[110]. It can be seen that the carbon concentration in austenite decreases with

increasing time, and the / interface moves to the right hand side, indicating

austenite growth and ferrite shrinkage.

Figure 3.7 Modelling of carbon distribution in austenite, employing the model in [71]

based on Fick‟s second law.

In general, carbon concentration can be well predicted by austenitization models

based on Fick‟s second law. However, most models neglect the incubation period,

therefore the starting temperature of austenitization cannot be calculated accurately.

Partial differential equations are widely used and depend greatly on both mathematical

theories and geometry of the models. Hence the equations are difficult to solve.

Therefore, a unified material model based on physical mechanisms is to be preferred.

Such a model which enables the prediction of start and end temperatures, as well as

the evolution of the volume fraction of austenite under different heating processes, will

be meaningful for the industrial practice.

3.3.2 Modelling of bainite transformation

There are two major approaches to model the kinetics of the bainite transformation:

nucleation controlled model and the diffusion controlled growth model of bainitic

laths[90].

3.3.2.1 Nucleation controlled model

The nucleation controlled model is based on the displacive and diffusionless nature

of the bainite transformation. The diffusionless growth of bainitic ferrite ceases when

the carbon concentration of the residual austenite reaches a value at which austenite

and bainite of identical chemistry have equal free energies. This dynamic limit can be

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8

Distance from the center of model symmetry (µm)

Carb

on c

oncentr

atino in

γ (%

)

t=0.2 s

t=0.6 s

t=2.0 s

t=6.0 s

t=30.0 s

Page 63: modelling of phase transformation in hot stamping of boron steel

63

represented by the plot of a temperature and carbon concentration curve. The limit

curve can be modified from the To curve (temperature vs. carbon concentration curve

when austenite and ferrite of identical chemistry have equal free energies) to allow for

the stored energy of bainite (~400 J/mol) [111]. A maximum volume fraction of bainitic

ferrite, max

Bv , is thus obtained. A normalised volume fraction of bainitic ferrite was

defined by Rees and Bhadeshia [112] as:

max

BB

nor B

vv

v 3.7

where Bv is the actual volume fraction of bainitic ferrite. Since the Widmanstätten

ferrite ( w ) and bainite originate from the same nucleus, the nucleation mechanisms of

w and bainite are considered to be identical, but the growth mechanisms are different.

A potential nucleus can develop into either bainite or w , depending on the available

driving force at the transformation temperature concerned. As a consequence, the

nucleation rate of bainite at any temperature, T, can be given as a multiple of the w

nucleation rate, wsI , at the start temperature of the w transformation, wsT :

1 2expB B

B w m Ns

ws ws

G GTI I

RTT R T T

3.8

where B

mG is the maximum possible free energy change on nucleation (a function of

the volume fraction of ferrite), B

NG is the value of the universal curve representing the

minimum necessary free energy change for displacive nucleation of ferrite at the wsT

temperature, 1 and 2 are empirical material constants. At the later stages in the

transformation, taking into account the „extended volume‟ [79], the normalised bainite

transformation rate is given as [112]:

max

max

1 1 exp expB B

B B B B B B Bnor avro nor nor norB

dv uI v v v v

dt v 3.9

where B

avru is the average volume of a single bainite subunit, B

oI is the initial value of

BI , B is the empirical autocatalysis factor.

3 0

B

mG

RT

3.10

Page 64: modelling of phase transformation in hot stamping of boron steel

64

01 2

B BB m N

ws ws

G GT

RTT R T T

3.11

where 0

B

mG is the initial value of B

mG , and 3 is an empirical material constant.

An analytical solution describing the relationship of time t, normalised volume

fraction B

norv and temperature T is given as:

32max 1 max

max

01 2

ln 1 ln 1 1

exp

BnorvB B B B B

nor norB B

B BB B m Navr o

ws ws

v v v v ev

tG GT

u IRTT R T T

3.12

where 1 , 2 , 3 are constants arising from the integration of Equation 3.9.

Material constants 1 , 2 , 3 , B and

B B

avr ou I can be determined by optimizing the

model with experimental bainite volume fraction vs. time curves at various

temperatures.

The above model was later improved by Rees and Bhadeshia [113], considering the

effect of carbon partitioning on free energy change, autocatalysis, the effect of

austenite grain size, as well as the change in bainite sub unit size with temperature and

was modified into the form as

1 02max

max 3

1 1 exp 1B B B

B B B B Bnor avr mnor nor norB

dv u K GKv v v v

dt v RT K

3.13

1 21B

Cx 3.14

1

1 1K LK

3.15

2 0

3

B B

m NK G G

K RT

3.16

where 1K , 2K , 3K , 1K , 1 and 2 are empirical material constants, Cx is the mean

carbon concentration, L is the parameter related to austenite grain size. The solution

of the equation is in the same form as that of the previous model, which is:

Page 65: modelling of phase transformation in hot stamping of boron steel

65

32max 1 max

max

2 021

3

ln 1 ln 1 1

exp

BnorvB B B B B

nor norB B

BB mavr

v v v v ev

tK GK

u KRT K RT

3.17

Determined material constants were given in [114]. The model was used to predict

isothermal transformation kinetics of Bainite for Fe-0.38C-1.29Si-1.73Mn (wt%) steel.

Reasonable agreement has been obtained, but the final volume fraction of bainite was

over predicted for temperatures greater than 400℃, as shown in Figure 3.8.

Figure 3.8 Comparison of predicted and measured volume fraction of bainite

transformation for Fe-0.38C-1.29Si-1.73Mn (wt%) steel [114].

The kinetic model developed by Bhadeshia is physical mechanism based, and can

be applied to isothermal bainite transformation. This has greatly assisted the

understanding and quantifying of the volume fraction of bainite transformation. The

overall kinetics are considered to be mainly controlled by the nucleation rate, rather

than the growth rate. However, the introduction of a thermodynamic limit accounting for

the stored energy of bainite, and variation of grain size effect with temperature and

steel composition make the model complicated to use.

3.3.2.2 Diffusion controlled model

According to the diffusionless mechanism for the formation of the nuclei, the driving

force for the transformation without composition change would increase with

decreasing temperature. However, it has been observed experimentally that the

nucleation rate of bainite decreases with decreasing temperature, which excludes the

displacive mechanism for nucleation [115, 116]. The bainite nucleation step is thus

Predicted

Volu

me fra

ction (

B%

)

Time (s)

Measured

Time (s)

Volu

me fra

ction (

B%

)

Measured

Predicted

Page 66: modelling of phase transformation in hot stamping of boron steel

66

treated as a diffusional process, suggesting the trend of temperature dependence is

mainly due to the diffusion coefficient rather than the driving force. Therefore, diffusion

of carbon in austenite plays an important role in determining the growth rate of bainitic

laths.

The classical nucleation theory is used to predict the nucleation rate of bainite

formation:

*

exp expb b

b b NQ GN K

RT RT

3.18

where bK is a material constant,

b

NQ is the activation energy. The exponential term

expb

NQ

RT

accounts for the diffusional process limiting the formation, and

*

expbG

RT

accounts for the work required to form the nucleus. *bG is the work

required to form a critical nucleus and can be evaluated according to the nucleation

model developed by Lange et al. [117].

Modelling of the growth of bainite takes into account the diffusion of carbon in

austenite and the presence of cementite precipitates left behind the growing front. The

growth rate is calculated according to the Trivedi equation for diffusional growth of plate

precipitates [118], and given as [115, 119]:

3

0 *

27

256

bb

b

c

Dv

3.19

where b

c is the critical curvature of a growing phase at the growth tip which gives the

maximum growth rate 0

bv , bD is a diffusivity coefficient given in [120], * is an

algebraic combination of the carbon supersaturation 0

b :

0*

2

0 0

2 11

2

b

b b

3.20

The evolution of bainite transformation fraction is modelled using the Kolmogorov-

Johnson-Mehl-Avrami equation [121-125]:

01 exp /b b b

eX V V 3.21

Page 67: modelling of phase transformation in hot stamping of boron steel

67

where b

eV is called the „extended‟ volume representing the volume filled by the grains

of the new phase without impingement, 0

bV is the initial volume to be transformed.

Under isothermal conditions, at the time period from t0 to t, the „extended volume‟

can be calculated by the integration of the volume of all the bainite laths nucleated

between t0 to t (number of laths×volume of a lath at t nucleated at a time between t0 to

t) over the concerned time period:

2

0 0

b b b b

eV t N v t t 3.22

where b is the multiplying factor adjusting the actual lath velocity. By substituting the

calculations of bN , 0

bv and b

eV into the Equation 3.21, the fraction of bainite can be

given as

22

1 0 01 exp expb

b b bcK QX t K v t t

RT

3.23

where 1

bK is a material constant which can be determined by optimizing the model with

experimental bainite volume fraction vs. time curves at various temperatures, 2

bK is the

ratio accounting for austenite grain boundary diffusion during bainite nucleation and its

value has been determined in [119], cQ is the activation energy for bulk diffusion of

carbon in austenite and is defined by [126].

The above model can be formulated easily to predict volume fraction of bainite

transformation for thermal processes. Under continuous cooling conditions, along the

cooling path from temperature Bs (start temperature of bainite transformation) to T, the

„extended volume‟ can be calculated by the integration of the volume of all the

nucleated bainite laths nucleated between Bs to T (number of laths×volume of a lath at

T nucleated at a temperature between Bs to T) over the concerned temperature range.

The formulation of volume fraction of bainite under continuous cooling conditions has

been given in [119].

Reasonable agreement has been achieved by using the diffusion controlled bainite

transformation model predicting volume fraction of bainite for both thermal and

isothermal cases. However, discrepancies exist in the prediction of upper bainite and

lower bainite. The diffusion controlled model is a more accurate description of upper

bainite. This may due to the fact that diffusion of carbon in lower bainite is mostly

limited within bainite laths rather than long range diffusion between austenite-ferrite

Page 68: modelling of phase transformation in hot stamping of boron steel

68

interfaces. It is also worth noting that in the case of the diffusion controlled growth

mode of bainite laths, solute drag effect has to be considered [127].

3.3.2.3 Modelling of strain effect on bainite transformation

Numerous discussions exist on the strain and stress effects of pre-deformation on

the bainite transformation [128-132]. External stress has been proved to accelerate

bainite transformation, especially at high transformation temperatures [128], and both

the ferrite and cementite components of the bainitic microstructure have been observed

to respond to stress [131]. It has been reported that the overall transformation kinetics

become slower and the maximum fraction of bainite obtained decreases in deformed

austenite [130], however, severe deformation results in an increased obtainable

maximum bainite fraction [129]. Experimental evidence from [132] shows that

deformation of austenite grains for low carbon steels increases bainite transformation

temperatures by approximately 100℃ which leads to the conclusion that the plastic

deformation of austenite grains may provide more energy to encourage the phase

transformations, and consequently shorten the incubation time.

Compared with the experimental studies of the stress/strain effect on bainite

transformation, modelling investigations in this area are limited in number. Based on

the thermodynamic principles of phase transformation [133], Garrett and Lin [63]

proposed a set of unified phase transformation constitutive equations to model the

incubation time and volume fraction growth of bainite, accounting for pre-deformation in

the austenite state.

It is given by thermodynamic principles that the driving force for a phase

transformation is the difference in value of Gibbs free energy between the parent phase

and product phase. If, assuming the new phase particle is spherical with a radius r, the

total free energy change is comprised of two parts for the new phase particle which can

be written as:

3 24

43

r V ABG r G r S 3.24

where VG is the decreased free energy per unit volume, and ABS is the increased

interfacial energy of small particles. It is given by Porter, Easterling and Doremus [134,

135] that

V V

eq

TG Q

T

3.25

Page 69: modelling of phase transformation in hot stamping of boron steel

69

where undercooling is T , eqT is the equilibrium temperature of austenite and bainite,

QV is the latent heat of fusion per unit volume. According to the experimental

observations in [132], Garrett and Lin [63] introduced the plastic deformation effect on

transformation by including the plastic strain energy density factor into the above

equation:

V V

eq

TG Q

T

3.26

where is the flow stress of the material during the deformation and is the

equivalent strain. Thus the critical radius of the nucleation new phase *r and the peak

value of the total free energy change *G (also known as the activation free energy

barrier), are deduced with consideration of the pre-deformation effect:

1*

2

eq

V eq

f T Tr

TQ f T

3.27

1.5 2

3*

2

4

eq

V eq

f T TG

TQ f T

3.28

where 1f , 2f , 3f and 4f are material constants. The growth rate of nucleated phase is

given in a form of a modified Gaussian type equation as:

*

5

expT T

r ff

3.29

where f is a function of , 5f is a material constant, *T is the temperature

corresponding to the shortest incubation time as given in [63]. The bainite volume

fraction rate fx is given as

1

2

* * *

6

7

exp 1 expf f

T T r r Gx f x

r RTf

3.30

where 6f , 7f , 1 , 2 are material constants.

The model can be determined by fitting with the experimental TTT diagram and the

volume fraction evolution curve of bainite transformation, and a close agreement has

been achieved [63].

Page 70: modelling of phase transformation in hot stamping of boron steel

70

The above model theoretically provides an advanced insight in modelling the strain

effect of pre-deformation on bainite transformation, and has enabled a very close

agreement in predicting the start temperatures of bainite transformation and the volume

fraction evolution of bainite transformation with strain effect to be achieved. However, it

is difficult to model the strain effect in practice merely by the introduction of the plastic

strain energy density factor. For example, in industry, the hot rolling of steel plates is

always followed by water quenching to produce multi phase steels. The hot rolled steel

plates remain in the air for a short time period t before water quenching. This period

allows for relaxation to take place. Meanwhile, bainite transformation may occur.

According to Garrett et al [63], during t , the external force is zero, therefore the

plates are stress free (internal thermal stress is usually ignored in this case), and the

plastic strain energy density factor is „zero‟. Thus, the pre-deformation (hot rolling

process) has no effect on bainite transformation of steels plates, which is not true. Thus

Equation 3.26 needs to be reviewed in calculating the strain effect on bainite

transformation.

In this thesis, an improved bainite transformation model is introduced based on

Garrett and Lin‟s model. Normalized dislocation density is introduced to account for the

strain effects on bainite transformation instead of plastic strain energy density factor.

The newly developed model can thus model the bainite transformation with strain

effects when the external force is zero, such as during the relaxation. Detailed

discussion is in Chapter 7.

Page 71: modelling of phase transformation in hot stamping of boron steel

71

Chapter 4

Modelling of austenitization

4.1 Introduction

One of the key features of this process is austenitization during heating and soaking

periods, for the purpose of forming a single austenite phase with a homogeneous

microstructure and small grain size. In addition to the quenching, controlling of the

austenitization is vital to obtain the required high strength of boron steel. The

characteristics of the austenite affect the component‟s microstructure and hence

mechanical properties resulting from the forming/quenching operations. Heating and

soaking for the austenitization need to be carefully designed so that the original phase

can be fully austenitized and the best mechanical property of austenite can be

achieved. In addition, the coating of the steel should be preserved and energy

consumption is minimised. Therefore, a thorough understanding of the austenitization

process is fundamental and this can be aided using a material model. Fick‟s second

law is often used to describe the austenitization process, however, it is difficult to solve

and the format is usually complex. Factors such as incubation period and various

heating characteristics are normally ignored. A mechanism based model which is easy

to solve and to implement would provide a more convenient method.

This chapter mainly deals with an experimental determination of alloy characteristics

for the development of a new model of austenitization, using unified constitutive

equations. The model is physical mechanism based, easy to integrate and convenient

to use.

4.2 Experimental investigations

In the hot stamping and cold die quenching process a work-piece is heated to a

temperature at which austenitization occurs. It is then soaked at this temperature, for

time to enable the amount of austenite to increase, before it is transferred to the

forming dies.

In industrial practice, engineers are more concerned with the effects of various

heating processes on austenitization of steels, so as to achieve either a full austenite

phase with lowest energy consumption, or to control the volume fraction of austenitized

Page 72: modelling of phase transformation in hot stamping of boron steel

72

material. To meet this requirement, a series of heat treatment tests were undertaken to

investigate austenitization of boron steel. Dilatometry was employed to study the phase

evolution during austenitization.

4.2.1 Experimental program

Austenitization of steel is diffusion controlled, hence it is strongly affected by the

heating process. A set of heat treatment tests were carried out using a Gleeble

simulator, to determine the effect of heating rate on austenitization characteristics. The

specific temperature-time profiles used are shown in Figure 4.1. The purpose of the

experiments is to investigate the effect of heating rate on austenitization of the boron

steel.

Figure 4.1 Test program and temperature profile for phase transformation study.

The material was heated to 950℃ at various rates. When the temperature is lower

than 600℃, a heating rate of 10℃/s was used for all the tests. This is because 10℃/s

is close to the heating rate in industrial heating practice and is lower than the maximum

heating rate required for boron steel, 12℃/s, protecting the Al-Si coating [50]. No phase

transformation is expected if T<600℃. When the test pieces were heated over 600℃,

various heating rates of 30℃/s, 10℃/s, 5℃/s, 1.25℃/s and 0.5℃/s were employed,

until the temperature reached 950℃. The material was soaked at 950℃ for 5 mins

followed by cooling at 30℃/s, which is higher than the critical martensite cooling rate,

27℃/s, of boron steel.

Time (sec)

30, 10, 5, 1.25,

and 0.5℃/s

10℃/s

Temp (℃)

30℃/s

950

600

5 mins

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73

4.2.2 Facilities and phase transformation measurement

A Gleeble 3500 simulator, by which temperature can be precisely controlled, was

employed for the heat treatment tests. It has a direct resistance heating system which

can heat specimens at rates of up to 10,000℃/s, and can hold steady-state equilibrium

temperatures. Thermocouples provide signals for accurate feedback control of

specimen temperatures. Employment of high thermal conductivity grips to hold the

specimen, allows accurate heating and cooling control. A quenching system enables

various cooling rates to be obtained. All the signals necessary to control thermal test

variables are recorded in real time, through the digital closed-loop thermal servo

systems.

The test piece was rectangular with a thickness of 1.5mm as shown in Figure 4.2. A

hole near each end was used for locating the test piece. A thermocouple was welded to

the centre of the specimen to control the temperature in the surrounding region.

Dilatation, due to phase transformation and thermal expansion, was measured using a

dilatometer at the midline of the specimen. In addition, thermal strain during the heating

and cooling process can also be measured.

Figure 4.2 Design of samples for heat treatment tests.

A typical strain vs. temperature curve is shown in Figure 4.3. When the temperature

rises from T to T1, the strain increases linearly, representing the thermal expansion of

the original phase of the steel. The slope is the thermal expansion coefficient of the

parent phase. When the temperature increases from T1 to T2, the strain evolution

deviates from a linear relationship with temperature, which indicates the occurrence of

phase transformation. This is due to the different lattice systems between the parent

phase, ferrite, and austenite. Ferrite is in a body-centred cubic (BCC) lattice, while

austenite has a face-centred cubic (FCC) lattice. For the same material, FCC has a

larger density than BCC. When a BCC system transforms into a FCC system, the

volume of a given mass of material will decrease, which leads to a drop of the dilatation

Page 74: modelling of phase transformation in hot stamping of boron steel

74

evolution rate or thermal strain rate. Consequently, the linear relationship of thermal

strain to temperature no longer exists. If the temperature continues rise above T2, the

thermal strain will increase linearly with temperature again, representing the thermal

expansion of austenite. At temperature T2, the parent phase has been fully

austenitized. The slope is the thermal expansion coefficient of austenite. Therefore

the starting temperature (T1) and finishing temperature (T2) of austenitization process

can be identified readily from the dilatation curves.

Figure 4.3 A typical strain-temperature curve obtained from heat treatment test.

The results of analysis of dilatation curves from each heat treatment test, are

summarized in Figure 4.4. Diamond symbols with the solid line represent the start

temperatures and square symbols with the solid line, finish temperatures.

Figure 4.4 Experimental start and finish temperatures for austenitization of the boron

steel, and volume fraction of austenite at heating rates of 1.25℃/s and 5℃/s.

0

4

8

12

0 200 400 600 800 1000

TEMP (℃)

Str

ain

10

E3

)

T T1 T2

αα βγ

complete

start

38%

95%

77%

21%

Page 75: modelling of phase transformation in hot stamping of boron steel

75

To highlight the phase transformation process, the enlarged strain-temperature

curves within a temperature range from 727℃ (1000K) to 950℃ (1223K) are presented

in Figure 4.5. It is shown in the figure that at a particular temperature, thermal strains

are smaller at lower heating rates, representing larger volume fractions of austenite.

The reason is in that, lower heating rates allow for longer times for phase

transformation. As a consequence, a larger amount of austenite is formed.

Figure 4.5 Strain-temperature curves within a temperature range from 727℃ (1000K)

to 950℃ (1223K) for tests under heating rates of 5℃/s, 10℃/s and 30℃/s, respectively.

In order to study the growing volume fraction of austenite at certain temperature

levels during continuous heating, heat treatment tests at two heating rates, with various

maximum temperatures values were designed and the heating processes are

illustrated in Figure 4.6. The samples were heated to 400℃ at a heating rate of 10℃/s.

For one test, after temperature achieved 400℃, the material was continuously heated

at 1.25℃ /s to 800℃ , and followed by quenching. For the other four tests, after

temperatures are above 400℃ , specimens were continuously heated at 5℃ /s, to

various temperatures, which are 700℃, 770℃, 840℃ and 900℃, and followed by

quenching. Specimens were quenched at a rate of 50 ℃ /s to obtain complete

transformation of the austenite to martensite. Hence, the volume fraction of martensite

in the final phase equals to the volume fraction of austenite immediately before

quenching. It was measured using SEM. Measured fractions of austenite (martensite)

are presented in Figure 4.4 by triangle symbols. Relevant SEM microstructures are

given in Appendix 1.

30℃/s

5℃/s

10℃/s

Page 76: modelling of phase transformation in hot stamping of boron steel

76

Figure 4.6 Test program and temperature profile for evaluation of austenite volume

fraction at different temperature levels.

It is shown from Figure 4.4 that for a heating rate of 5℃/s, when the percentage of

austenite increases from 0 to 21%, the temperature increase is around 24℃; when the

percentage of austenite increases from 77% to 95%, the temperature increase is

around 60℃. The austenitization rate becomes smaller as the absolute amount of

austenite increases. It is thus concluded that, the austenite evolution rate is also a

function of volume fraction of austenite itself, and decreases with the growing of

austenite fraction. To summarise, the evolution of austenite is multi-controlled by

diffusion coefficient, temperature, heating rate and accumulation of austenite.

4.3 Modelling of austenitization

The modelling technique of austenitization is discussed in this section. A set of

mechanism-based unified constitutive equations for modelling the evolution of

austenite during continuous heating was developed. The effects of temperature and

heating rate on incubation time were rationalized, and their effects on the development

of austenite volume fraction were analyzed as well. The evolution of strain with the

increasing volume fraction of austenite under different heating rates was evaluated.

Optimization techniques and procedures were developed to determine the model from

dilatometry experimental results.

4.3.1 Modelling of incubation time

Incubation time, which is a function of temperature and heating rate, is an important

parameter in determining the starting temperature of austenitization. Higher heating

Time (sec)

5°C/s

50°C /s

700,770,840,900°C 800°C

1.25°C /s

10°C /s

Temp (°C)

400°C

Page 77: modelling of phase transformation in hot stamping of boron steel

77

rate leads to a higher start temperature. A variable x, which varies from 0 to 1, was

introduced to describe the incubation period. As soon as the temperature rises to the

critical austenitization temperature of the steel, usually 727℃, incubation starts. At the

beginning of the incubation period x=0.0. When the incubation period approaches the

end, the value of x is close to 1.0, and austenitization process occurs. At a higher

heating rate, incubation covers a wider temperature range above 727℃. To model the

incubation process, the following rate equation is introduced

11 1

c

Tx A x T

T 4.1

where A is a material constant related to the incubation period, γ1 is the constant to

regulate heating rate during the incubation period. T is absolute temperature, Tc is the

theoretical critical austenitization temperature of the steel, Tc=1000K=727℃ . If the

temperature is lower than Tc, 1c

T

T

<0, then 0x .

4.3.2 Modelling of the evolution of the austenite fraction

As the austenitization of steel is a diffusion process, it is necessary to include a

diffusion factor in the model to evaluate the volume fraction of austenite. The diffusion

coefficient, which is temperature dependent for a thermal austenitization process, can

be described as

0 exp

DQD D

RT 4.2

where D0 is a material constant, QD is the activation energy, T is the absolute

temperature and R is the universal gas constant.

The introduction of D represents the effect of temperature on diffusion rate of carbon

or alloy in steels. It is also known from the experimental results that austenitization

takes place after incubation and the phase transformation is fast at the beginning. With

the accumulation of austenite, the austenitization process slows down. The growing

rate of the fraction of austenite is a function of accumulated volume fraction of

austenite. Thereby, the evolution rate of volume fraction of austenite is multi-controlled

by several parameters,

, , ,a av f D T T v 4.3

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78

Both temperature and heating rate have to be included because austenitization of

steel is temperature dominated; the heating process plays an important role, as can be

seen from Figure 4.5. To model the volume fraction of austenite evolution, the following

rate equation is introduced

21 1

a

a

m

n

a a

c

Tv D x v T

T

4.4

where ma, na are material constants used to regulate the effects of temperature and

accumulation of austenite; 2 is the constant to regulate the effect of heating rate

during the phase transformation. The term 1

c

Tx

T is introduced to determine the

start temperature of austenitization process. If 1 0c

Tx

T

, then 0av .

Austenitization only occurs when 1 0c

Tx

T

.

4.3.3 Evaluation of dilatometry strain

Throughout the process of austenitization, the volume change of the material is

characterized to evaluate the amount of austenite that has been transformed. The

thermal strain can be summarized from the phase transformation part of a dilatometric

strain curve (as shown in Figure 4.5), which is

a v av T v 4.5

where (K-1) and (K-1) are thermal expansion coefficients of ferrite and austenite,

respectively (shown in Figure 4.3), which can be measured from the dilatometry strain

curve directly. For boron steel, 15.2516 6e /K, and 21.6305 6e /K. v is

the material constant related to volume difference between original material lattice,

ferrite, and the austenite lattice.

4.3.4 The unified constitutive equations

By introducing the incubation variable and implementing the austenite volume

fraction evolution equation (Equation 4.4) into the thermal strain equation (Equation

4.5), a set of unified constitutive equations modelling the austenitization process are

formulated as follows:

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79

11 1c

Tx A x T

T

21 1

a

a

m

n

a a

c

Tv D x v T

T

4.6

a v av T v

T T T t

where A, 2 ,

1 , ma, na, and v are material constants. and are thermal

expansion coefficients of ferrite and austenite, respectively. D is the diffusion coefficient

and is temperature dependant. Temperature T is in Kelvin. Tc is the theoretical critical

austenitization temperature of the steel, where Tc=727℃=1000K. If 1 0c

T

T

, then

0x ; if 1 0c

Tx

T

, then 0av .

4.4 Determination and validation of the unified material

model for austenitization

The modelling work is based on the experimental dilatometry results obtained from

the heat treatment tests. The model is calibrated via the optimization method by fitting

the predicted dilation curves from the model to the experimental data. The optimization

technique is based on minimizing the sum of the squares of the errors between the

experimental and computed data. It utilises a „combined methods‟ approach with both

Fast Evolutionary Programming (FEP) and Gradient-Based Method (GBM) techniques

to search for the material constants within given boundary ranges [136]. An objective

function is used to compare the fit with experimental data [137, 138], and best fits with

corresponding material constants were selected. The unified constitutive equations

were then integrated using a specific integration code written in C++, with the selected

material constants. Finer adjusting of material constants maybe applied during the

fitting, depending on the agreement level of integrated curves and experimental data.

The material constants can be finalized after best fitting result was achieved.

The solid curves in Figure 4.7 are the fitted curves, derived by the computations.

Symbols are the collection of data from dilatometry experiments. Determined material

constants are listed in Table 4.1. By the determined model, the start and finish

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80

temperatures of austenitization under heating rates of 5℃/s, 10℃/s and 30℃/s are

predicted and summarized in Figure 4.8 by solid lines. From the figure, some

discrepancy can be observed during phase transformation process, but the computed

temperature features follow well with the trend of experimental data (demonstrated by

solid triangles). Possible sources for errors may include:

Precision of experimental data: experimental data were collected by a dilatometer

with high accuracy; however the dilatometer may not have been placed in the

absolute middle of a sample. In addition, the location of thermal couples as well

as the location of maximum temperatures may be deviated from the mid line of

specimens, which brought in the additional experimental error. However, the

precision of experimental data has been improved by repeating the tests.

Effectiveness of the optimisation software: restrictions in the optimization

techniques, such as pre-selected boundary conditions, formulations of objective

functions, may bring in errors.

Accuracy of the modelling laws: there will always be a trade off between

mathematical simplicity and modelling accuracy, which leads to the discrepancy

of fitting curves from experimental data.

Figure 4.7 Comparison of experimental (symbols) and computed (solid curves) thermal

strain-temperature relationships for the phase transformation parts from dilatometry

curves. Heating rate varies from 5℃/s to 30℃/s.

To investigate the reliability of the model, further computations were carried out to

predict start and finish temperatures of austenitization, as well as the evolution of

volume fraction of austenite under heating rates of 0.5℃/s and 1.25℃/s. Results are

also summarized in Figure 4.8 by solid lines. Predicted volume fraction of austenite

under certain temperature levels are indicated by the star symbols. Figure 4.9 is a

5℃/s

10℃/s

30℃/s

Page 81: modelling of phase transformation in hot stamping of boron steel

81

comparison of experimental (symbols) and computed (solid curves) results of the

volume fraction of austenite against temperature, at different heating rates. It is shown

from Figure 4.8 and Figure 4.9 that the predicted curves exhibit the same trend as the

experimental data, showing that the proposed unified austenitization model, together

with the material constants listed in Table 4.1, can describe the growing fraction of

austenite, as well as the starting and finishing temperatures of austenitization under

various heating processes accurately, the austenitization process of the boron steel is

well represented by the model.

Table 4.1 Material constants determined for Equation set 4.6.

A (K-1·s-1) D0 (s-1) na (-) θv (-)

296.7 6900.0 1.7 0.00279

ma (-) QD (J·mol-1) γ1 (-) γ2 (-)

1.77 68000.0 0.1 0.82

Figure 4.8 Comparison of experimental (by triangles) and computed (by solid lines and

starts) starting and finishing temperatures of austenitization process, as well as the

evolution of fraction of austenite under heating rates of 1.25℃/s and 5℃/s.

Since the model is physical mechanism based, and does not rely on the parent or

austenite microstructures, it can be potentially used for various types of steels, with the

re-determined material constants for the specific steel. However, the model cannot

predict microstructure evolution during austenitization, or the carbon or alloys diffusion

Austenitization TEMP-Heating rate

700

800

900

1000

0.1 1 10 100

Heating rate(℃/s)

Te

mp

era

ture

(℃) d

f

start

95%

38%

77%

21%

finish

Page 82: modelling of phase transformation in hot stamping of boron steel

82

behaviour, nor the homogenization process of austenitization of steels which is also

diffusion controlled.

Figure 4.9 Evolution of experimental (symbols) and computed (solid curves) fraction of

austenite vs. temperature. Heating rates are 1.25℃/s, 5 ℃/s and 30℃/s.

4.5 Summary

(1) The austenitization process is diffusion controlled and therefore a temperature

dominated process. Modelling techniques of austenitization based on Fick‟s second

law are effective in predicting the carbon diffusion process but difficult to solve. A

unified material model in the format of constitutive equations of austenitization based

on physical mechanisms is necessary and meaningful for industrial practice.

(2) Dilatometry is an effective method to study the phase transformation of the steel.

The starting and finishing temperatures of austenitization can be evaluated from

dilatation curves. For continuous heating processes, the starting and finishing

temperatures are higher when the heating rate is higher, and they are lower when the

heating rate is lower. The developing rate of austenite becomes smaller with the

accumulation of austenite.

(3) A series of unified constitutive equations are proposed to describe the

austenitization process under various heating routes. Data from dilatometry

experiments of boron steel were employed for optimization and determination of the

equations. The model is physical mechanism based and has the potential for being

applied to various steels. Effects of heating rate on incubation time and effects of time

and temperature on development of austenite during heating were accounted for.

Incubation time and volume fraction of austenite under different heating processes can

1.25℃/s

5℃/s

30℃/s

5℃/s

1.25℃/s

Page 83: modelling of phase transformation in hot stamping of boron steel

83

be predicted. Evolution of the thermal strain during phase transformation can also be

rationalized by the model combined with dilatation curves.

(4) The model is easy to be integrated, and it does not rely on the parent phase or

the microstructure of the material.

Page 84: modelling of phase transformation in hot stamping of boron steel

84

Chapter 5

Thermo-mechanical testing

5.1 Introduction

Thermo-mechanical testing is widely performed for assessing the effects of

deformation at elevated temperature, on material properties, and the results can be

used for understanding material behaviour at temperature and for selecting materials

for engineering applications.

Uniaxial hot tensile tests were conducted to simulate the real industrial conditions

experienced in hot stamping and cold die quenching of steel sheet. Test-pieces were

heated under the specific heating route identical to that which is employed by

ArcelorMittal, and heated to the specified preheat temperature for a time period equal

to that of a furnace soaking commensurate with the hot-working operation.

Subsequently, the test-pieces were cooled to a determined temperature and deformed

under tension. Test results in terms of stress-strain curves and anisotropy are

summarised.

A formability test was introduced to study the formability of boron steel. Different

failure modes were observed when the forming speed was different. Thickness

distribution at the test-piece section along radial direction was measured. The testing

results will be used to compare with FE analysis in the later chapters.

5.2 Uniaxial hot tensile tests

5.2.1 Aim

The objective of performing these uniaxial hot tensile tests was to study and

simulate the deformation behaviour of boron steel under similar conditions to those

existing in a real HSCDQ forming process, and to determine the mechanical properties

of the boron steel subsequent to thermo-mechanical deformation. The stress-strain

relationship obtained from tensile tests at different temperatures was used to develop

the viscoplastic-damage model of boron steel.

Page 85: modelling of phase transformation in hot stamping of boron steel

85

5.2.2 Equipment and testing procedure

In this work, a Gleeble material simulator was employed for tensile testing (Figure

5.1). Thermocouples or an optional infrared pyrometer provide signals for accurate

feedback control of specimen temperatures. In Figure 5.1 b), one end of the

thermocouple is connected to the Gleeble feed back system as shown on the top of the

figure, the other end is welded on a tensile sheet test-piece. The test-piece is clamped

between the anvils. C-Gauge (linear variable differential transformer (LVDT) measuring

the lateral dimension change at central of a tensile test-piece) is clamped at the centre

of the tensile test-piece. By using a C-Gauge, instantaneous cross-sectional area as

well as strain in the lateral direction where the C-Gauge is clamped can be monitored

and recorded. Thermal and mechanical parameters, i.e. time, temperature, C-Gauge

values, and force were monitored and recorded throughout deformation. True stress

was calculated from the instantaneous force and C-Gauge values while true strain was

deduced from the C-Gauge data and cross-sectional dimensions of test-pieces.

a) b)

Figure 5.1 Gleeble 3800 system used for hot-tension. (a) The control console with a

Digital Closed-Loop Control System and the load unit with test chamber are shown in

the picture. (b) Gleeble test unit showing attached thermocouple wires and C-Gauge.

The testing programme and temperature profile of the tests is shown in Figure 5.2. T

is temperature. The test-piece was firstly heated to 400℃ at 10℃/s, then heated to

700℃ at 5℃/s followed by continuous heating to 925℃ at 1.25℃/s. This particular

heating route is used by ArcelorMittal for its industrial stamping process. The test-piece

was then soaked at 925℃ for 1 minute to ensure the steel was fully transformed into

austenite. Determination of soaking time and temperature was based on the results in

Chapter 4. After soaking, the test-pieces were cooled to testing temperatures at a

cooling rate of 50℃/s, which is similar to that experienced in transporting the blank to

C-Gauge

Thermocouple

Test chamber

Control console

Page 86: modelling of phase transformation in hot stamping of boron steel

86

the press, in the industrial process. A holding time of 5 seconds at the forming

temperature was introduced prior to tensile testing, to compensate over-cooling after

the sharp temperature decrease at 50℃/s, to obtain a stable, uniform temperature.

Natural cooling was applied after tension tests.

In order to test viscoplastic behaviour of the boron steel, tensile tests were carried

out at various strain rates, ranging from 0.01/s to 10.0/s, which is sufficient to cover the

strain rate ( ) range experience in industrial practice. Although the nominal forming

temperature in industry is 700℃, temperature will change as heat is lost from blank to

dies, therefore tests were conducted over a range of forming temperatures between

600℃-800℃. Strain rates and temperatures of tensile tests (under the ticked conditions)

are listed in Table 5.1.

Figure 5.2 Temperature profile and testing procedure of hot tension tests.

Table 5.1 Strain rates and temperatures of tension tests at different temperatures.

έ (s-1)

T (℃) 0.01 0.1 1.0 10.0

600 √

700 √ √ √ √

800 √

5.2.3 Test-piece design

Previous research work has shown that test-piece geometry plays an important role

in the determination of hot tensile characteristics [139, 140]. The shape of a typical

Time (s)

Isothermal tensile test: Strain to failure; Strain rate: 0.01, 0.1, 1.0

& 10.0s-1

, T = 600-800℃

Natural cooling

Heating rate

10℃/s

5℃/s

1.25℃

/s

700

400

925 Soaking time: 1min

Hold at forming temperature for 5s

T (℃)

Cool to testing

temperatures at 50℃/s

Page 87: modelling of phase transformation in hot stamping of boron steel

87

tensile test-piece, cut from sheet, is shown in Figure 5.3. It has enlarged ends for

gripping. The cross-section area of the gauge section is reduced relative to that of

shoulders so that deformation and failure can be localized in this region. The effective

gauge length is the length over which elongation is measured and is centred within the

reduced section. The distances between the ends of the gauge section and the

shoulders should be long enough so that the shoulders do not constrain deformation,

and the effective gauge length should be large relative to its width, otherwise, the

stress state will not be uniaxial. Standard dimensions of tensile test-pieces are

published by ASTM, ISO, JIS, and DIN, but they are not identical.

Figure 5.3 A tensile test-piece showing the gauge length, effective gauge length and

diameter.

There are various ways of gripping the test-piece depending on the shape of the

ends[141], but the most important concern is to ensure that the test-piece can be held

at the maximum load without slippage or failure in the grip section, and bending should

be minimized at the same time. In this research, the sheet test-pieces gripped with

jaws and located with pins, were employed.

5.2.4 Temperature distribution in test-pieces of different gauge

length

Due to the use of resistance heating on the Gleeble, test-piece temperature is not

uniform along the gauge length but is highest midway between the grips. Theoretically,

the test-pieces should fracture in the hottest region perpendicular to the test-piece axis,

where a thermocouple should be placed. The longitudinal thermal gradient is quite

important in assessing whether tensile test can achieve isothermal conditions. Test-

Shoulder

Effective gauge length

Gauge length

Width

Page 88: modelling of phase transformation in hot stamping of boron steel

88

pieces of different gauge lengths were heated following a predefined temperature

history and the longitudinal temperature distribution were measured for each test-piece.

Figure 5.4 shows the dimensions of the tensile test-piece employed for the hot

tensile tests. The width of the sample cross the gauge section is 12mm. L0 (length of

reduced section) of 14mm, 30mm, and 50mm, were used. The thickness of the steel

sheet was 1.5mm, as provided by ArcelorMittal. The testing procedure is described in

Figure 5.5. Four pairs of thermocouples were welded on each test-piece, equi-spaced

along half the gauge length, as shown in Figure 5.6 a). Thermocouple TC1 was used

by the Gleeble controller to ensure the defined temperature history (Fig 5.5) was

followed at the centre of the test-piece. Thermocouples TC2, TC3 and TC4 measured

temperatures locally. It was assumed that the temperature distribution in the other half

of the test-piece was a mirror image of the measured one. The temperatures measured

during the soaking process of test-pieces with different gauge length are shown in

Figure 5.6 b). Hot zones observed after tensile testing are shown in Figure 5.7 It is

seen that when the gauge length is shorter, the temperature gradient is much larger in

the central area; hence tensile deformation will have greater possibility to be localized

at the centre. Since the tensile test with a uniform temperature distribution on the test-

piece is preferred in this research to simulate an isothermal tension, gauge length of

50mm was employed for uniaxial tension described in later sections. Based on the

temperature distribution in Figure 5.6, effective gauge length of the test-piece with

gauge length of 50mm is chosen as 26mm. The reason is the lowest temperature

within the effective gauge length area is around 727℃ which is the theoretical

austenitization temperature of steel. Only the material within this area can be

austenitized during the soaking. Strain to failure of each sample was tested by uniaxial

tension tests conducted at 700℃, and at strain rates of 0.1, 1.0 and 10/s. Results are

shown in Appendix 2. It is shown from the results that a larger gauge length leads to

larger strain to failure.

Figure 5.4 Dimensions of tensile test-pieces for hot tensile tests. Gauge lengths (L0),

14mm, 30mm and 50mm.

Page 89: modelling of phase transformation in hot stamping of boron steel

89

Figure 5.5 Schematic illustration of testing procedure.

a)

b)

Figure 5.6 a) Locations of thermocouples welded on test-pieces. b) Temperature

distribution along test-pieces of gauge lengths 14, 30 and 50mm.

Time (s)

Cooling rate 50 ℃/s

Heating rate 10℃/s

5℃/s

1.25℃

/s

700

400

925 Soaking time: 1min

Temp (℃)

Room temperature

Temp distribution

0

200

400

600

800

1000

-20 -15 -10 -5 0 5 10 15 20

distance to central (mm)

Te

mp

(℃

)

14mm

30mm

50mm

TC1 TC2

TC3

TC4

TC1

TC3 TC2

TC4

Page 90: modelling of phase transformation in hot stamping of boron steel

90

Figure 5.7 Hot zones in test-pieces of different length.

5.2.5 Anisotropy at hot stamping conditions

Anisotropy describes the phenomena of chemical bond strengths being directionally

dependent. In polycrystalline materials, it may due to certain texture patterns which are

often produced during manufacturing of the material. In the case of rolling, as a result

of the crystallographic structure, "stringers" of texture are produced in the direction of

rolling, which can lead to vastly different properties in the rolling and transverse

directions. Sheet metals thus generally exhibit anisotropic mechanical properties.

Anisotropic behavior is especially important in processes such as stamping and deep-

drawing. Elastic anisotropy is usually described by young‟s modulus [142, 143]; plastic

anisotropy is characterized by r value or planar anisotropy [144].

5.2.5.1 Definition of anisotropy

If tensile tests are performed on test-pieces cut from sheet material at different

orientations to the prior rolling direction, there maybe different mechanical responses of

the material due to anisotropy. The parameters that are widely used to characterize the

anisotropy of sheet metal are strain ratio or r-value, and the planar anisotropy.

The r value is defined as the ratio of width strain, εw, over thickness strain, εt,

measured in a tensile test before necking occurs:

/w tr 5.1

where 0ln /w w w , and 0ln /t t t . w0 and w are the initial width and the

instantaneous measure of width, respectively. t0 and t are the initial thickness and the

instantaneous measure of thickness, respectively. The value of r equals 1 for an

isotropic material.

Page 91: modelling of phase transformation in hot stamping of boron steel

91

Since the value of r greatly depends on the direction, relative to that of rolling, along

which the test-pieces are cut, an average r-value, r , is more often quoted to describe

the anisotropy of a material. r is defined as:

0 90 452

4

r r rr

5.2

The r value reflects the difference between plastic properties parallel and normal to

the plane of the sheet, and hence named normal anisotropy. In a tension case, when

r <1, deformation in the thickness direction is easier than in the tension direction, the

thickness decreases after deformation; when r >1, deformation in thickness direction

is more difficult and thickness increases after tension.

Planar anisotropy describes the degree of anisotropy in the plane of the sheet, and

is defined as:

0 90 452

4

r r rr

5.3

A material will show almost isotropic behaviour if r close to zero.

5.2.5.2 Anisotropy of the tested material

The normal anisotropy of boron steel was reported by Turetta and Merklein (2007),

as shown in Figure 5.8. From the results it can be seen that the effect of strain rate is

less than that of temperature. The steel exhibits significant normal anisotropy at

temperatures less than 650℃, but is almost isotropic at 800℃, indicating the thickness

of a blank sheet will decrease sharply when being deformed in tension at below 650℃.

Figure 5.8 Average normal anisotropy (without being austenitized) sensitivity to forming

temperature and strain rate (test-pieces were cooled at 50K/s after austenitization)

[145].

Page 92: modelling of phase transformation in hot stamping of boron steel

92

Since planar anisotropy represents directional variation of mechanical properties in

the plane of a metal sheet, the stress strain relationship of uniaxial tension can be

employed to investigate the planar anisotropy of boron steel. Five groups of tests using

a Gleeble 3500 were performed. All tests were repeated three times to confirm the

features observed. The test-pieces were cut with their axes at angles of either 0°, 45°

or 90° to the rolling direction (Figure 5.9). They were heated to 925 ℃ and kept at this

temperature for 5 minutes to ensure the material was fully austenitized (Figure 5.10),

and were subsequently cooled at 50℃/s to the tensile testing temperature. Since

temperature has a much stronger effect than strain rate on anisotropy, and most of the

hot stamping is expected to occur in a temperature range of 600℃ ~ 800℃, the

isothermal tension tests to failure were performed under a strain rate of 0.1s-1 and at

two forming temperatures; 600℃ and 800℃.

Figure 5.9 Samples were cut along the directions of 0°, 45° and 90° to the rolling

direction.

Figure 5.10 Test programme and temperature profile.

Time (s)

Isothermal tensile test: Strain to failure; Strain rate:

0.1s-1

, T = 600 & 800℃

Air cooling

Cool to testing temperature

at 50℃/s

Heating rate

10℃/s

925 Soaking time: 5min Temp (℃)

90°

45°

0° Rolling direction

Page 93: modelling of phase transformation in hot stamping of boron steel

93

Stress-train curves are plotted in Figure 5.11. It can be seen that, no significant

difference exists between the properties of test-pieces cut at 0o and 90o, at either

testing temperature. These results correspond to those of Turetta and Merklein [145,

146], for this steel composition and is due to the fact that the steel was fully

austenitized after being soaked at 925℃ for 5 minutes. The nucleated austenite

replaced the original anisotropy structure from the rolled steel.

a)

b)

Figure 5.11 Stress-strain relationships for the samples along various directions (0°, 45°

and 90°) and formed at (a) 600℃ and (b) 800℃.

Temperature=800℃

0

50

100

150

200

250

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Strain

Str

es

s(M

Pa

) 0°

90°

Temperature=600℃

0

50

100

150

200

250

300

350

400

0 0.05 0.1 0.15 0.2 0.25 0.3

Strain

Str

es

s(M

Pa

)

90° 45°

Page 94: modelling of phase transformation in hot stamping of boron steel

94

5.2.6 Mechanical behaviour

5.2.6.1 Effect of austenitization conditions

The austenitization condition affects the microstructure of the steel before testing

and consequently affects the mechanical property of the material. Therefore, the effect

of the austenitization conditions on the tensile tests was investigated. Relevant

microstructures of the material after tension were observed using SEM.

Tensile tests were carried out on the Gleeble 3500 using the dog-bone shape

samples. The effect of austenitization conditions on flow stress was investigated for the

temperature range from 870 ℃ to 950℃, and soaking time varying from 1 to 10

minutes. The heating procedure was the same as that used by ArcelorMittal in

industrial practice. Tensile tests to failure were conducted at 700℃ under a strain rate

of 0.1s-1 (Figure 5.12). The soaking conditions were planned as shown in Table 5.2.

Figure 5.12 Test programme and temperature profile.

Table 5.2 Soaking time and temperatures of tests.

T (℃)

Time (min) 870 925 950

1 √

5 √

10 √

Time (s)

Isothermal tension test: Strain to

failure; Strain rate: 0.1s-1

, T = 700℃

Air cooling

Cool to testing

temperature at 50℃/s

Heating rate

10℃/s

5℃/s

1.25℃

/s

700

400

870/925/950 Soaking time: 1, 5, and 10min

Temp (℃)

Page 95: modelling of phase transformation in hot stamping of boron steel

95

Each test was repeated twice to confirm the features observed. The stress strain

curves of all tests are given in Appendix 3. Variation of stress at a strain level of 0.1

between different test-pieces is plotted in Figure 5.13, and strain to failure is shown in

Figure 5.14. It is seen from the result that stresses measured at a strain level of 0.1

from stress-strain curves are identical for all soaking conditions and strain to failure

decreases with increased value of soaking temperature and soaking time. This is due

to austenite grain growth and incomplete austenite transformation for the 1 minute

soaking time at 870℃. This is shown in the micrographs of test-pieces after tension, in

Figure 5.15. The boron steel was fully austenitized after being soaked at 925℃ for 5

minutes, and the microstructure became coarser at higher temperature and for longer

soaking time, hence ductility decreases. A very small volume fraction of ferrite

remained in the test-piece soaked at 870℃ for only 1 minute, which contributed to the

relatively high ductility. However, the effect is not significant for the industrial hot

stamping process. In general, when the soaking condition is between 870℃ and 950℃

for 1 to 10 minutes, which covers conditions in industrial practice, the soaking

temperature and time do not have significant effect on flow stress, and have limited

effect on strain to failure of the material. Hence, for this work the austenitization

condition was fixed at 925℃ for 1 minute for characterising mechanical properties.

Figure 5.13 Stresses at a strain level of 0.1 for test-pieces subjected to different

austenitization conditions (870℃ for 1min, 925℃ for 5 min, and 950℃ for 10 min).

stress values when strain=0.1

100

140

180

220

260

300

860 880 900 920 940 960

Soaking temperature (℃)

Str

es

s(M

Pa

)

Page 96: modelling of phase transformation in hot stamping of boron steel

96

strain to failure-soaking Temp

0.1

0.15

0.2

0.25

0.3

0.35

860 880 900 920 940 960

Soaking temperature (℃)

Str

ain

to

fa

ilu

re

Figure 5.14 Strain to failure of test-pieces subjected to different austenitization

conditions (870℃ for 1min, 925℃ for 5min, and 950℃ for 10min).

Figure 5.15 Microstructures of test-pieces after tension. Soaking conditions were 870℃

for 1min, 925℃ for 5min, and 950℃ for 10min.

5.2.6.2 Mechanical characterisation

The effects of temperature and strain rate on ductility of the material were

investigated for the temperature range from 600℃ to 800℃, and strain rates from

0.01s-1 to 10.0s-1. The testing program and conditions are shown in Figure 5.2 and

Table 5.1. Stress-strain curves of the hot tensile tests are given in Appendix 4. Strain to

failure and maximum stress of each test was compared, and the results are

summarised in Figure 5.16.

Time (s)

50℃/s

Heating rate

10℃/s

5℃/s

1.25℃

/s

700

400

950 Soaking time: 10min Temp (℃)

925

870 1min

5min

Air cooling

Page 97: modelling of phase transformation in hot stamping of boron steel

97

a)

b)

Figure 5.16 a) Strain to failure and b) maximum flow stress versus temperature for hot

tension tests at different strain rates (0.01/s, 0.1/s, 1.0/s and 10.0/s).

It is shown by the tests that 1) at 700℃ , the strain to failure decreases with

increasing strain rate and the maximum flow stress increases with increasing strain

rate; 2) at 0.1/s, the strain to failure increases with increasing forming temperature and

the maximum flow stress decreases with the increasing forming temperature.

Viscoplastic behaviour of the material at 700℃ is significant. Empirically, viscoplastic

behaviour of steel becomes more significant as the temperature increases. The reason

is that time dependent softening mechanisms, such as recrystallization and recovery

Strain to failure-Temperature

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

550 600 650 700 750 800

Temperature(℃)

Str

ain

to

fa

ilu

re

Strain rate=0.1/s

Strain rate=1.0/s

Strain rate=10.0/s

strain rate=0.01/s

0.01/s

0.1/s

1.0/s

10.0/s

Stress (max)-Temperature

100

150

200

250

300

350

400

450

500

450 500 550 600 650 700 750 800

Temperature (℃)

Ma

xim

um

flo

w s

tre

ss

(M

Pa

)

Strain rate=0.1/s

Strain rate=1.0/s

Strain rate=10.0/s

strain rate=0.01/s

10.0/s

1.0/s

0.1/s

0.01/s

Page 98: modelling of phase transformation in hot stamping of boron steel

98

become more pronounced at high temperatures. Further discussion of this matter is

given in Chapter 6.

The obtained stress-strain relationship in Figure 5.16 and Appendix 4 is employed to

develop and validate the visco-plastic damage model of boron steel, which is

introduced in Chapter 6.

5.3 Formability tests

5.3.1 Equipment and testing procedure

Formability testing was introduced to assess the formability of boron steel. Figure

5.17 a) shows the apparatus used. A 3-D drawing die set is given in Figure 5.17 b).

The punch has a diameter of 80 mm; the blankholder has an inner diameter of 84 mm

and outer diameter of 136 mm. The temperature history and testing procedure are

shown in Figure 5.18. The test-piece was firstly heated in the heating furnace (No. 1 in

Figure 5.17 a)). Soaking temperature was 925℃. A soaking time of 3 minutes was

chosen to ensure the material was fully austenitized before forming. Temperature was

monitored by the thermal couple welded to the test-piece. The austenitized test-piece

was then fast transferred from heating furnace to the die set (placed on the top of the

blankholder plate) for stamping. Transfer time was within 3 to 5 seconds to avoid

significant temperature drop of the test-piece. The forming temperature was measured

by thermal couple and was around 700℃. Ram speed was controlled by setting the

servo-hydraulic system on the press. (No. 4 in Figure 5.17 a)). During the deformation,

the top plate was pushed down at a constant speed by the hydraulic press. The top

plate first contacted with the test-piece which was placed on the blankholder plate, and

then the top plate, the test-piece and the blankholder plate moved down together until

the test-piece was stamped by the punch which was supported by the base plate and

the gas spring. The test-piece was tightly clamped between the top plate and the

blankholder plate during the deformation to avoid material flow. Consequently, the

central hole on the test-piece was expanded by the punch. After stamping, the formed

test-piece was held in the closed dies until the temperature dropped to room

temperature. Two ram speeds, 15mm/s and 100mm/s, were selected. A high speed

camera (No. 2 in Figure 5.17 a)) with assistance of a triple prism placed above the die

set was employed to record the whole forming process.

Page 99: modelling of phase transformation in hot stamping of boron steel

99

a) b)

Figure 5.17 a) Apparatus of formability tests: 1- heating furnace, 2- high speed camera,

3- forming die set, 4- 25 tonne ESH high speed press. b) A 3-D drawing of the die set.

Figure 5.18 Temperature profile and testing procedure of formability test.

5.3.2 Test-piece design

The test-piece design is shown in Figure 5.19. The unit is millimetre. The thickness

of the steel sheet was 1.5mm. Four alignment holes, 5.5mm in diameter, are equi-

distributed on a pitch circle, 150mm in diameter. A central hole, 16 mm in diameter,

Top plate

Blankholder plate

Punch

Stop ring

Base plate

Gas spring Punch support

Die support

Rig base plate

1

2

3

4

Time (s)

Ram speeds are 15mm/s and 100mm/s

Cooling in the die to room temperature

Transfer to forming die set

Heating rate

<10℃/s

925 Soaking time: 3 mins

Temp (℃)

Page 100: modelling of phase transformation in hot stamping of boron steel

100

was cut in the centre of the test-piece, for the qualitative evaluation of plastic

deformation.

Figure 5.19 Test-piece dimensions for formability test.

5.3.3 Mechanical response of the material

Test-pieces after stamping are shown in Figure 5.20. It can be seen that, the central

hole is significantly larger for the test at a high ram speed (100mm/s) compared with

that of a low ram speed (15mm/s). Failure modes are a circumferential crack at a low

ram speed (Figure 5.20 a)), and a radial crack at a high ram speed (Figure 5.20 b)),

respectively. This is because the uneven temperature distribution, consequently there

was an uneven hardness distribution in the test-pieces, during stamping. Since the

punch was spherical and the test-piece was flat, contact of the punch to the test-piece

firstly occurred near the region around the central hole. For the test with a ram speed of

15mm/s, the material around the central hole had been cooled down before significant

deformation occurred, thus the material near the central hole was hardened. Material

away from central hole was thus softer than that near the central hole. Material at the

outer edge was cooled by the blank holder and therefore, was hardened as well. As a

consequence, localized deformation and failure took place in the transitional region

from the centre to the edge of the deformation zone, forming a circumferential crack.

For the test with a ram speed of 100mm/s, the movement of the punch is too fast to

4x Ø5.5

Ø16.0

Ø150.0

170

170

Page 101: modelling of phase transformation in hot stamping of boron steel

101

allow sufficient time for the material in contact with the punch to cool significantly; thus

hardening of the material near the central hole was not significant, which kept the

material relatively warm and easy to deform. Material at the outer edge was cooled by

the blankholder and hardened, therefore deformation was localized near the central

hole, resulting in radial cracks propagating from its edge. By measuring the central hole

diameters as a function of ram stroke, plastic deformation of the test-piece can be

quantitatively evaluated.

a) b)

Figure 5.20 Hot stamped test-pieces. Ram speeds are a) 15 mm/s, b) 100 mm/s.

Evolution of the diameter of the central hole, d, and ram stoke s were recorded by

the high speed camera. The diameter of the central hole was normalized by dividing it

by its original value; that is dt/d0, where dt/d0>1. The relationship of dt/d0 vs. st is

summarized in Figure 5.21. Only the data, which were measured before cracking

occurred, were accounted for. It is shown in the figure that the normalized central hole

diameter increases faster for the test at 100mm/s. For a given ram stroke, the

normalised central hole diameter of the test at 100mm/s is larger than that of the test at

15mm/s.

dt dt

Circumferential crack mode Radial crack

mode

Page 102: modelling of phase transformation in hot stamping of boron steel

102

Figure 5.21 Normalized diameter dt/d0 vs. ram stoke st. Ram speeds are 15mm/s and

100mm/s.

The thickness distribution of the deformed test-pieces was measured. Small pieces

were cut from the deformed test-pieces, as shown by dashed lines in Figure 5.20. The

thickness (t) of deformed regions of cut pieces against the X-Position, as shown in

Figure 5.22, was measured; results are summarized in Figure 5.23. Thickness of

deformed regions was normalised by t/t0, where t0 is the original thickness of steel

sheets. Punch strokes during deformation for the tests at ram speeds of 15mm/s and

100mm/s were pre-set as 25mm. Deviation existed due to the experimental error. Ram

speeds for the tests shown in Figure 5.22 were a) 15mm/s and b) 100mm/s

respectively. Thinning areas which indicate the localized deformation can be obviously

seen from the figures.

a) b)

Figure 5.22 Sectional views of deformed regions which were cut from test-pieces. Ram

speeds are a) 15mm/s and b) 100mm/s.

X

Y

X

Y

Thinning area

Thinning area

15mm/s

100mm/s

Page 103: modelling of phase transformation in hot stamping of boron steel

103

Figure 5.23 Thickness distributions against X-Position.

A viscoplastic damage model for boron steel was developed (described in Chapter 6)

and integrated with commercial FE software, ABAQUS. FE analysis of the formability

test was carried out and is described in Chapter 7. The results observed in Figure 5.22

and Figure 5.23 are compared with the results of FE analysis in Chapter 7, to verify the

viscoplastic damage model.

100mm/s

15mm/s

Page 104: modelling of phase transformation in hot stamping of boron steel

104

Chapter 6

Development of unified viscoplastic

damage constitutive equations

6.1 Introduction

In this chapter, a set of unified viscoplastic damage model is formulated to describe

the mechanical properties of hot stamped boron steel. The equations predict material

behaviour based on experimental observation associated with thermodynamic insight.

The evolution rate of normalized dislocation density, isotropic hardening and damage

are accounted for. Stress strain curves obtained from uniaxial tension tests are

employed to calibrate the equations by means of computer based optimisation and

integration techniques.

6.2 Development of viscoplastic constitutive equations

It has been described in Chapter 5 that boron steel exhibits significant viscoplastic

behaviour when formed at elevated temperatures. To interpret the temperature and

strain rate dependency of stress, dislocation slip is regarded as having a key role.

Dislocation glide is the main mechanism in a deformation process. The increased

dislocation density due to plastic deformation raises the resistance of a metallic

material, and the material is thus work hardened [147]. Many observations have shown

that flow stress is proportional to the square root of dislocation density in pure metals

[148]. However, recovery, in the forms of dislocation climb and grain-boundary

migration occurs simultaneously with deformation and tends to decrease dislocation

density. It is well known that dislocations in deformed pure metals aggregate to form a

cell structure and at high temperature the cell wall is adjusted to from a sub-boundary

[149]. This results in a large internal stress component in the metal matrix, and

therefore the effect of recovery on flow stress cannot be neglected. In general, the

driving force for recovery increases as the internal stress (dislocation density)

increases. Therefore, the recovery rate increases as work hardening proceeds and

finally they are balanced to realize a steady-state and the flow stress saturates. As

temperature rises, the recovery rate also increases. As a result, the strain to the

steady-state decreases and the saturated flow stress decreases [149]. The change of

Page 105: modelling of phase transformation in hot stamping of boron steel

105

dislocation density with the deformation history is crucial in describing the deformation

mechanism of boron steel. In addition to dislocation density, in order to identify the flow

rule, a work hardening law has to be specified.

6.2.1 Flow rule

The flow rule describes the stress-strain relationship of a material when it is

deformed, incorporating factors such as, initial yield stress, material hardening,

dislocations, and creep.

The total strain T can be divided into the reversible or elastic strain

e and the

irreversible or inelastic strain p . p can be further separated into the components

anelastic, plastic, and viscoplastic strain. This partition is based on the physics of the

phenomena as elastic deformation essentially corresponds to variation in the

interatomic distances without changes of place, while plastic deformation implies „slip‟

movements with modification of interatomic bonds. By Hooke‟s law,

T pE 6.1

Young‟s modulus is formulated as:

0 exp EQ

E ERT

6.2

where E0, Young‟s modulus, and QE, activation energy, are material constants.

The decoupling of elastic and plastic effects can be justified on the basis of the

physics and chemistry of solids, thermodynamics, and phenomenological experiments.

Since the material exhibits viscoplastic behaviour, plastic strain is a general term given

to both time-independent and time-dependant inelastic material straining.

The relation existing between plastic strain rate p and stress can be described by

Norton‟s law [150]:

**( / )N

p 6.3

where * and

*N are coefficients which are characteristic of each material and

temperature. The value of the exponent *N and consequently that of

* depends on

the range of rate considered: for high rates, the geometric effect of the reduction in the

cross-sectional area of a tension specimen becomes dominant, thus the values

obtained for *N are generally higher. The exponent has a tendency to decrease for

low rates [151].

Page 106: modelling of phase transformation in hot stamping of boron steel

106

Other elementary facts may be derived from special creep tests and relaxation tests.

Creep tests at several levels of stress can be used to make a clear choice between two

commonly accepted hardening hypotheses [152], strain-hardening or time-hardening,

to represent primary creep. Experiments performed on different materials show a

general agreement with the hypothesis of strain hardening. Relaxation tests

characterize viscosity and can be used to determine the relation between stress and

rate of viscoplastic strain. Each point on a relaxation curve gives the stress and rate of

viscoplastic strain /p E [151]. For each of the three kinds of test, relaxation,

creep and hardening, the evolution of p and p can be calculated and represented

together in a ( p , p ) graph with stress as the parameter, as shown in Figure 6.1,

which proves experimentally that a mechanical law of state:

,p pf 6.4

can be defined in the domain of variation of p and p .

Figure 6.1 Representation of the three basic tests in the space of the variables p and

p (after Lemaitre et al. [151]).

Taking Norton‟s law and the affinity of hardening curves into account, a reasonable

specification of the function f consists in taking it as the product of power functions. The

viscosity-hardening law can therefore be written as [151, 153]:

* *1 2* M M

p pK 6.5

where *K ,

*

1M and *

2M are material constants which are temperature and material

dependent. Rearranging the above equation, a viscoplastic flow rule can be formed as:

*2*

1

*

mm

p

pK

6.6

Page 107: modelling of phase transformation in hot stamping of boron steel

107

where *

1m

p can be classified as the strain hardening parameter. *

1m is the hardening

exponent which varies approximately between 2 and 50; *

2m is the viscosity exponent

with a value of the order of 2 for very viscous materials and of 100 for materials of low

viscosity which warrant a plasticity law; *K is the coefficient of resistance, which for

metals, varies between 100 and 10000MPa [151]. The classical strain hardening law is

empirically determined. It neglects the initial yield of a material, and is incapable of

predicting softening. The stress dissipation potential of material flow (with hardening)

can therefore be expressed as a function of e H k [151, 154, 155], where

e is

the equivalent stress, H internal stress due to material hardening, and k is the initial

elastic limit, or initial material yield stress. Consequently, the flow rule can be

formulated as:

cn

ep

H k

K

6.7

where nc is the viscosity exponent. If nc has a value from 1 to10, the material exhibits

significant viscoplasticity and if nc>=20, the material behaviour is close to perfectly

plastic deformation.

It has been concluded by Dunne et al. [156] that the initial yield stress k decreases

with increasing temperature as the external energy required to overcome obstacles to

plastic strain (such as atomic bonds) decrease. Therefore, it can be formulated by the

Arrhenius-type equation:

0 exp

Qk k

RT

6.8

where Q is the activation energy associated with initial plastic slip, k0 is a material

constant and R is the gas constant. In addition, both K and exponent nc are

temperature dependent parameters, and generally increase with decreasing

temperature [151]. They are given by:

0 exp

QK K

RT

6.9

0 exp n

c

Qn n

RT

6.10

in which K0 and n0 are material constants, Q and Qn are activation energy parameters

for K and nc respectively. Empirically, nc various dramatically with temperature: nc has a

Page 108: modelling of phase transformation in hot stamping of boron steel

108

higher value when temperature is low, and a lower value when temperature is high. For

instance, typically for steels, if T=500℃, nc ranges from 15 to 20; if T=700-800℃, nc

=2~5; and if T=900~1000℃, nc =2~4.

Inverting Equation 6.7, the equivalent stress can be given as:

1

cn

e pk H K 6.11

It is necessary to specify the form of the hardening function H and this will be

discussed later. The constitutive relation clearly shows the decomposition of the stress

as the sum of a hardening term, H, which represents the current size of the elastic

domain, and a viscosity term, v , often called the viscous stress:

1

cn

v p pK 6.12

The decomposition of stress dissipation is illustrated in Figure 6.2.

Figure 6.2 Schematic illustration of decomposition of stress dissipation.

6.2.2 Dislocation density evolution

The net developing rate of dislocations for a microstructure before dynamic/static

recrystallization takes place is contributed by two parts: trapping of dislocations during

plastic slip and concurrent dislocation annihilation during the recovery process [157].

Dislocations are accumulated when mobile dislocations are trapped and halted at

discontinuities, such as at grain boundaries, inclusions or when mobile dislocations

interact with other dislocations during the plastic deformation. The storage of

dislocations was modelled by Nes [158] and Krausz [159] using the Kocks-Mecking

approach:

Page 109: modelling of phase transformation in hot stamping of boron steel

109

*

pM

bL

6.13

where p is the true plastic strain rate; M* is the constant defining the texture of the

polycrystalline structure; L is the dislocation slip length, over which a dislocation can

run; b is Burger‟s vector.

With increasing density, the stored energy of dislocations provides the driving force

for static recovery which is time dependent. It dominates at elevated temperatures

when annealing effectively removes dislocations from the matrix [155, 157]. Static

recovery occurs faster when dislocation density is higher. The time dependency of

static recovery has been formulated by Sandstrom and Lagneborg [160, 161] in the

form:

* * 22static M 6.14

where *M is the mobility of dislocation recovery and

* is the average energy per unit

length of dislocation. Taking the recovery of dislocation into account, the dislocation

density rate can be described as [160, 161]:

*

* * 22pM

MbL

6.15

The first term in the above equation describes the development of dislocation structure

and the second represents the static recovery due to annealing at high temperatures.

Dynamic recovery is due to continuous reorganisation of dislocations during

straining, by dislocation cross-slip (at low temperatures) or dislocation climb (at high

temperatures), which has weak time dependency [155, 162, 163]. A dynamic recovery

based dislocation density evolution equation was introduced by Estrin [162] as:

* *

1 2

p

dk k

d

6.16

where the coefficient *

1k characterizes the process of dislocation storage, and *

2k

represents a thermally activated process of dynamic recovery by dislocation cross-slip

or dislocation climb. Hence, the dynamic recovery is directly proportional to the

dislocation density [162]:

*

2

dynamic

p

k

6.17

Page 110: modelling of phase transformation in hot stamping of boron steel

110

Equations 6.15 and 6.17 enable dislocation accumulation and both dynamic and

static recovery to be modelled. During a deformation process, dislocation density will

develop to a stable state balanced between dislocation accumulation from plastic

deformation and concurrent dislocation annihilation by recovery. This is a complex

phenomenon comprising both dynamic and static recovery mechanisms.

In practice, the absolute dislocation density of a material, , is extremely difficult to

determine, therefore, a normalized dislocation density is employed in this thesis. The

normalized dislocation density was defined by Lin et al. [153, 164] as:

01

6.18

where 0 is the initial dislocation density and the dislocation density in deformed

materials. For annealed boron steel, the dislocation density in the undeformed

austenite phase is 0 . When 0 , 0 . With an increasing level of deformation,

dislocation density will increase, until recovery or recrystallization stabilise or reduce

the amount of dislocations. There always exists a practical limit to the amount of

dislocations that can be sustained within the material, denoted as max . This is far

greater than 0 . When max , 0 max/ 0 , 1 . Therefore, the normalised

dislocation density varies from 0, representing the initial fully annealed state, to 1,

corresponding to the saturated state of a dislocation network after severe deformation.

By this definition, the absolute values of 0 and max are no longer required when

describing the dislocation state of a material.

In consideration of high temperature deformation mechanisms, static and dynamic

recovery, a constitutive equation for normalized dislocation density evolution is

proposed as:

11 pA C 6.19

where A and 1 are material constants, and C is a temperature dependent parameter:

0 exp CQ

C CRT

6.20

in which C0 is a material constant, QC is the activation energy for static recovery. The

first term in the Equation 6.19 represents the development of dislocation density due to

plastic strain and dynamic recovery of the dislocation density, which is similar to

Equation 6.16, apart from the fact that dislocation storage rate is simplified to linearly

increase with plastic strain rate. The dynamic recovery term enables the normalized

Page 111: modelling of phase transformation in hot stamping of boron steel

111

dislocation density to be limited to the saturated state of a dislocation network when

1 [157]. The second term in Equation 6.19 models the effect of static recovery on

the evolution of dislocation density during high temperature annealing. This expression

comes from Equation 6.14. For simplicity, the parameter C is introduced to represent

the combined effects of the mobility of grain boundary (*M ) and the average energy

per unit length of a dislocation ( * ). The exponent 1 replaced the power 2 in Equation

6.19, providing higher flexibility of the equation, so that it is suitable for a wider range of

materials and deformation conditions [157]. Equation 6.19 is valid for microstructures

without recrystallization.

6.2.3 Isotropic work hardening

Plastic straining occurs by the formation and movement of dislocations. The material

is thus work hardened due to part of the mobile dislocations remaining locked in the

material lattice and interacting with other dislocations, forming new obstacles to

dislocation slip [165]. From Nes‟s study [158], the strain required to active dislocation

slip is remarkably affected the dislocation slip length L . The density of dislocations

increases the slip length L and a higher stress level is required to induce plastic

deformation. It is found that the mean dislocation slip length is governed by the inverse

square of the dislocation density,1/ 2

[158]. By using the normalised dislocation

density , the isotropic hardening effect is defined as [166]:

H B 6.21

where B is a temperature dependent parameter.

It is observed in the stress strain curves obtained from uniaxial tension tests,

described in Chapter 5, that work hardening in boron steel is slightly more significant

than in other types of steel [155, 167]. The isotropic hardening effect of boron steel can

thus be modified as:

0.4H B 6.22

and B is given by

0 exp

QB B

RT

6.23

Where B0 is a material constant, and Q is the activation energy. This hardening law

describes not only the isotropic hardening caused by the stacking and interaction of

Page 112: modelling of phase transformation in hot stamping of boron steel

112

dislocations but also allows for material softening resulting from reduction of dislocation

density during relaxation.

6.2.4 Uniaxial unified viscoplastic constitutive equations

From sections 6.2.1~6.2.3, the dislocation based uniaxial unified viscoplastic

constitutive equations were developed, as given in Equation set 6.24, to describe the

flow rules of a material without damage, in which isotropic hardening and evolution of

dislocation density were taken into account. The Isotropic hardening parameter H is a

function of dislocation density, which varies with plastic strain and recovery.

1

0

0

0.4

1, 0

1, 0

1

cn

p

p

T p

H k

K

when

when

A C

H B

E

6.24

Relevant temperature dependent constants are given by:

0

0

0

0

0

0

exp

exp

exp

exp

exp

exp

nc

C

E

Qk k

RT

QK K

RT

Qn n

RT

QC C

RT

QB B

RT

QE E

RT

6.25

T is the absolute temperature. The constants can be determined from experimental

data using optimisation techniques. It is worth noting that the material constants k, K,

and B are related with dislocation movement inside of the material, therefore, the

activation energy Q is employed for these constants to simplify the model. Reference

values of Q can be found in [155,166,167].

Page 113: modelling of phase transformation in hot stamping of boron steel

113

6.3 Development of viscoplastic damage equations

Modelling of damage is necessary for the prediction of formability and failure in

metals under various loading conditions. The failure of a metal is assumed to involve

the degradation of structure which is complex due to the nucleation, growth and

coalescence of defects, such as microvoids and microcracks [168, 169]. Damage

mechanisms vary with a number of factors, for example the loading process and

forming temperature, hence different models are required corresponding to each

damage mechanism. Damage variables are introduced to reflect average material

degradation at a macromechanics scale, and thus continuum damage mechanics

(CDM) was developed.

A representative quantity of research work on CDM has been presented in books by

Rabotnov [152], and Lemaitre and Chaboche [170]. In the described theories, it is

assumed that if the value of a specific damage variable reaches a certain level, the

material cannot sustain the applied load anymore and failure occurs. This concept has

been used to study the failure of materials under high temperature creep [171], creep-

cyclic plasticity [172], and formability in metal forming [173].

It is evident that the main reason for the damage in metal forming is ductile fracture

or intergranular fracture occurring in the crystalline materials [168, 174]. In cold forming,

multiplication of dislocations is the dominant deformation mechanism and only modest

grain boundary sliding occurs [175, 176]. Therefore, nucleated voids are usually close

to second phases within grains. Once a microvoid has been nucleated in a plastically

deforming material matrix, by either debonding or cracking of a second-phase or

inclusion, the stress-free surface of the void causes a localized stress and strain

concentration in the adjacent plastic field [177]. With further plastic flow of the matrix,

the void will thus undergo volumetric growth and shape change, and tend to form

macroscopic cavities or cracks leading to macroscopic fracture [178]. Early work on the

development of damage models was based on the modelling of the rate of change in

the radius of the void in the principal directions of strain rates for the remote strain rate

field [179]. The average rate of void growth in a unit cell is given by Li et al. [175], and

the growth of the non-spherical void is given by Boyer et al. [180]. The void growth

model can be integrated with a yield function to model the shrinkage of yield surface of

materials in plastic deformation.

In hot forming conditions, i.e. when the deformation temperature is greater than half

the melting temperature (Tm), damage development is believed to be associated with

intergranular damage [181] which corresponds to the nucleation of cavities at grain

Page 114: modelling of phase transformation in hot stamping of boron steel

114

boundaries [182]. Grain boundary cavity formation is a kinetic phenomenon. At low

stress levels, the damage is void-like [183]. This damage mechanism normally exists in

long-term high temperature creep, and results in very low ductility of a material [184]. A

model of this type of damage evolution has been given by Dyson [185] and Lin et al.

[186]. At high stress levels, the voids may link to form grain boundary cracks [183]. A

void nucleates at grain boundaries, and then grows larger during creep by diffusion of

atoms away from it, or by the plastic flow of the material surrounding it, or by a

combination of both [187]. When failure is incipient, that is when damage is large, this

is the dominating damage mechanism. A model has been developed by Cocks and

Ashby [188] to predict void growth rate based on grain boundary diffusion mechanisms.

In superplastic deformation, damage arises at grain boundaries as well. This is

because the dominant deformation mechanism is grain rotation and grain boundary

sliding. When the deformation is so large that stresses at grain boundaries are not

relaxed sufficiently, cavities nucleate [189]. These cavities can grow larger by different

mechanisms, for example, stress-directed vacancy diffusion [190], superplastic

diffusion [191], and plastic deformation [192]. Their void volume fraction increases

linearly with strain. Damage laws for superplastic materials have been developed by

Lin et al. [193] and [194].

In the work undertaken by this author, the metal is formed at a relatively high

temperature and stress state so that the damage mechanisms arising are different from

those described above. Hence a particular damage model was developed to represent

damage evolution in hot stamping of boron steel.

6.3.1 Damage evolution

Under a hot metal forming condition when the forming temperature is normally half

Tm above, and strain rates are higher than 1.0 s-1, plasticity induced damage may

occur around inclusions and hard particles. The grain boundary diffusion at high

temperatures which facilitates grain rotation and grain boundary sliding tend to be

inhibited, due to the high strain rates and small grain size of the original material.

It has been shown experimentally that microdamage occurs at both grain

boundaries and around second phases; more damage was observed at grain

boundaries for low strain rate tests and a significant amount of voids was observed

around second phase regions in high strain rate tests [184]. It is shown that the

plasticity – induced damage (around inclusions or hard particles) is similar to the

damage observed in cold metal forming [155], but the damage around grain boundaries

is different from the „creep - type‟ and „superplastic - type‟ damage. The reason is that

Page 115: modelling of phase transformation in hot stamping of boron steel

115

in hot forming, there is less time allowing grain rotation, grain boundary diffusion or

grain boundary sliding. Instead of multiple voids, macrowedge cracking arises at grain

boundaries. It is believed that grain boundaries are weakened by diffusion and

microcracks form under the deformation stress. The damage dominating in hot forming

conditions can be schematically illustrated, as shown in Figure6.3.

Figure 6.3 Schematic illustration of damage mechanism under hot metal forming

condition, after Lin et al. [184].

In cold metal forming, damage is normally plasticity induced, and resulting voids

nucleate around second phases which are mostly within grains [195]. In hot

deformation, higher strain rates and consequent higher flow stresses result in more

microcracks; thus damage is a function of flow stress and plastic strain rate p ,

given by:

, ,e pD f T 6.26

where D is the damage evolution rate, T is the absolute temperature. Odquist and

Hult[196] have further demonstrated that the damage rate can be a separable function

of , T and D, therefore the above equation was present as

d e pD f D 6.27

where d is a material constant to represent the temperature effects on damage

evolution.

In the case of high temperature creep at a high stress level, damage occurs in the

form of voids or cracks accumulating on grain boundaries. A void can grow by power –

law creep of the surrounding matrix, especially towards the end of life, when the

damage is significant. A model has been developed by Cocks and Ashby on damage

growth rate due to continuum cavity growth as [188]:

Hot deformation

Particles

Holes and cracks

Page 116: modelling of phase transformation in hot stamping of boron steel

116

11

1d

cg cg mn

cg

D DD

6.28

where cgD is the damage due to continuum cavity growth, as a function of void radius

voidr and the spacing of growing voids voidl given by, 2 2/cg void voidD r l , nd is a constant,

and m is the minimum creep rate. From the model, it can be determined that the

damage evolution rate increases with the development of damage itself. It has been

found from experimentally derived void volume% versus strain curves for simple

tension, that damage development increases from nucleation to failure by a factor of

1

1 cgD [191]. This phenomenon is similar to that which occurred in boron steel

during the uniaxial tension tests, which is obviously exhibited by stress-strain curves.

After necking was initiated, failure developed at an increasing rate. The factor

2

1

1f D

D

is employed to rationalise the significant damage growth at the end

stage of hot tension tests. The power 2 provides flexibility to the damage model so

that it can be used for various materials and deformation conditions.

Based on the damage mechanism presented in Figure 6.3 and the two models

discussed above, a new damage model suitable for hot deformation material is

proposed as:

21

e p

dDD

6.29

The value of D lies in a range from 0 to 1. When D=0, the material is undamaged. A

value of D=1, would represent a completely disintegrated material, which is practically

impossible, as metals fail at damage levels lower than D=1. The temperature

dependent parameter d can be given as:

0 exp D

d

Q

RT

6.30

where DQ is a material constant related to the activation energy of damage.

Page 117: modelling of phase transformation in hot stamping of boron steel

117

6.3.2 Modification of the material flow stress rule

The flow rule has to be modified to account for the damage term in the stress-strain

relationship. Based on the continuum creep damage mechanics, the evolution of

damage with time causes an accelerating creep rate and failure. To account for this

phenomenon which is due to the imposed stress being sustained by a decreasing

amount of sound material as damage accumulates, the concept of nett stress has been

proposed as a state variable. Thus nominal stress σ is replaced by nett stress, σnett

where, / 1nett D and 0 1D [197]. Thus the material is considered to be

subjected to an augmented stress over the undamaged area 1-D and this nett stress

drives further deformation and damage. By this reasoning, the flow rule for uniaxial

case is formulated as:

1

0

0

11

1

1, 0

1, 0

cn

p

H kD

K D

when

when

6.31

The factor 1

1

1 D

provides the equation with higher flexibility.

6.4 Uniaxial viscoplastic damage constitutive

equations

The intention of the author of this work is to develop a set of unified viscoplastic

constitutive equations to model the evolution of dislocation density, hardening, damage

and to rationalise their relationships and effects on viscoplastic flow of boron steel. The

mechanism based unified viscoplastic damage equations for hot metal forming have

the form:

Page 118: modelling of phase transformation in hot stamping of boron steel

118

1

2

3

0

0

0.4

11

1

1, 0

1, 0

1

1

1

cn

p

p

p

d

T p

H kD

K D

when

when

A C

H B

DD

E D

6.32

The temperature dependent constants are given by:

0

0

0

0

0

0

0

exp

exp

exp

exp

exp

exp

exp

nc

C

E

Dd

Qk k

RT

QK K

RT

Qn n

RT

QC C

RT

QB B

RT

QE E

RT

Q

RT

6.33

The material constants within the constitutive equation set have been determined

from experimental data using an optimisation technique. The particular numerical

integration code was programmed using reverse Euler method to integrate the

equations.

6.5 Determination of the equations via optimisation

The equation set has a total of 16 material constants that must be determined.

These are: k0 , K0 , n0 , B0 , E0 , A , C0 , Q , Qc , Qn , QD, QE , γ1, γ2, γ3, 0 . They are

determined by fitting theoretical and experimental stress strain curves for different

Page 119: modelling of phase transformation in hot stamping of boron steel

119

strain rates and forming temperatures. The tension flow stress data are presented in

Chapter 5.

The values of the material constants were determined by the optimization

techniques developed in-house [136-138]. The unified constitutive equations were then

integrated using a specific integration code written in C++, for fine adjustment of the

constants. Through trial and error and with a detailed understanding of the physical

base of each constant, best fit values were obtained incrementally.

Figure 6.4 is a comparison between experimental and fitted material curves for

uniaxial hot tension tests. The solid curves are the fitted curves derived by the

computations. Symbols are the experimental stress strain data given in Chapter 5 and

Appendix 4. Computed values of the material constants are listed in Table 6.1. From

the figure, some discrepancy can be observed. The flow stress at a large strain level

and small strain rate is under estimated, but the main features, such as Young‟s

modulus, maximum flow stress, and strain to failure, follow the trend of experimental

data well. Possible sources for the errors may come from:

Modelling laws: the employed equations may need to be improved by further

understanding of the physical deformation mechanism of boron steel, so that the

model can describe the material behaviour in all the details needed, to account

for the features observed.

Inaccurate experimental data: the data has been generated at high temperatures,

where inhomogeneity of deformation due to uneven temperature distribution of

the test-piece, is most likely to adversely affect results. In addition, the location of

thermal couples and C Gauge may have deviated from the centre-line of some

samples and not have been precisely in the highest temperature region.

Page 120: modelling of phase transformation in hot stamping of boron steel

120

a)

b)

Figure 6.4 Comparison of experimental (symbols) and computed (solid curves) stress

strain curves. a) Forming temperature is 700℃, strain rate is from 0.01/s to10/s; b)

Forming temperature is from 600℃ to 800℃, strain rate is 0.1/s.

diff Strain rate-700C

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

strain

str

ess(M

Pa)

700-0.01

700-0.1

700-1.0

700-10.0

comp

0.1/s

1/s

10/s

0.01/s

diff Temp-0.1/s

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

strain

str

ess (

MP

a)

600-0.1

700-0.1

800-0.1

comp

600℃

700℃

800℃

Page 121: modelling of phase transformation in hot stamping of boron steel

121

Table 6.1 Material constants determined for Equation set 6.32 and 6.33.

6.6 Summary

In this chapter, a viscoplastic-damage model is introduced consisting of a set of

unified constitutive equations. The model is able to predict the mechanical behaviour of

boron steel under a range of forming temperatures and strain rates. The evolution of

work hardening, normalised dislocation density, and damage can be described. The

developed model was embedded in the commercial FE software ABAQUS. FE

analysis was carried out using the developed model, which is described in Chapter 7.

Q (J/mol)

Qc (J/mol)

Qn (J/mol)

k0 (MPa)

B0 (MPa)

C0

(-) β0 (-)

E0 (MPa)

8400 99900 10650 12.4 80 55500 1.39e-4 1100

K0 (MPa)

n0 (-)

γ1 (-)

A (-)

γ2 (-)

γ3 (-)

QD (J/mol)

QE (J/mol)

30 0.0068 3.1 5.222 1.54 17.5 50000 17500

Page 122: modelling of phase transformation in hot stamping of boron steel

122

Chapter 7

Investigation of phase

transformation during cooling

7.1 Introduction

Quantitative research results have shown that pre-deformed austenite may have an

influence on the subsequent bainite transformation [63, 128-131, 198]. It is thus

expected that the preliminary deformation will affect the bainite transformation in terms

of changing the start and end of bainite transformation temperatures or the volume

fraction of the material that has been transformed. In other words, pre-deformation

changes the CCT diagram of boron steel.

In this chapter, experimental investigation and modelling of phase transformation

during cooling were studied. The interest lies in the bainite and martensite

transformation as these are the primary phases in hot stamped components. A model

to predict bainite transformation, considering the effect of pre-deformation, has been

developed and validated. The volume fraction of martensite during quenching can thus

be easily calculated as only bainite and martensite transformation were obtained during

quenching. The proposed model has been proved to agree well with experimental

results.

7.2 Experimental investigation of phase transformation

under cooling processes

Tests were conducted using a Gleeble 3800 to study the bainite and martensite

transformation during the cooling process. The material was fully austenitized before

being deformed and quenched. Start and end temperatures of bainite and martensite

transformations, and the fraction of transformed phase were investigated.

7.2.1 Phase transformation without pre-deformation

The steel was heat treated as described in Figure 7.1. A constant heating rate of

10 ℃ /s was chosen. The material was quenched after it had been thoroughly

austenitized at 950℃ for 5 mins. Cooling rates of quenching were varied from 10℃/s to

Page 123: modelling of phase transformation in hot stamping of boron steel

123

50℃/s. Based on the CCT diagram of boron steel, only bainite and martensite can be

obtained when a cooling rate is higher than 10℃/s. The specimens employed for the

tests were the same as those described in Chapter 4, Figure 4.2.

Figure 7.1 Test program and temperature profile for phase transformation study.

The dilatation curves obtained from the tests during cooling have been normalized

into thermal strain vs. temperature, which are size independent, as shown in Figure 7.2.

In Figure 7.2 a), cooling rates are 10℃/s, 20℃/s and 25℃/s. In Figure 7.2 b), cooling

rates are 30℃/s and 50℃/s. It can be seen from Figure 7.2 that when the cooling rates

are 10℃/s and 20℃/s, the dilatation strain deviates from a linear relationship with

temperature when T is around 600℃. This means bainite is transformed. In addition,

the further the strain curve is away from the linear decrease relationship with the

temperature, the more fraction of bainite is transformed. Therefore, the fraction of

bainite obtained decreases with increasing cooling rate. When the cooling rate is

greater than 25℃/s (Figure 7.2 b), the thermal strain decreased almost linearly with

decreasing temperature when T>400℃ (above the martensite transformation). It

means no bainite transformation occurred during cooling. When T is around 400℃,

there is a sharp change in the linear relation between the dilatation strain and

temperature which marks the start of martensite transformation. Therefore, a single

phase of martensite is thus expected in the final phase.

Time (sec)

10, 20, 25, 30

and 50℃/s

10℃/s

Temp (℃) 950℃

Page 124: modelling of phase transformation in hot stamping of boron steel

124

a)

b)

Figure 7.2 Dilation curves obtained for phase transformation tests presented in Figure

7.1, when cooling rates varies from 10℃/s to 50℃/s.

It is possible to deduce the start temperatures of bainite and martensite

transformation from dilatation curves, using the same method as described in Chapter

4. According to the dilatation curves, a new phase, either bainite or martensite starts to

form when the linear relationship of thermal strain and temperature changes during the

cooling process. The results are summarized in Figure 7.3. The continuous solid curve

is the CCT diagram of the boron steel provided by the steel supplier ArcelorMittal.

Symbols present the experimentally measured start temperatures of bainite and

martensite transformation. It is shown from the figure that, the experimental results

coincide with the CCT diagram, when no deformation was introduced before quenching.

The martensite transformation temperature is approximately 400℃, and the nose of the

bainite transformation temperature is approximately 520℃.

Strain-Temp

0

2

4

6

8

10

12

100 200 300 400 500 600 700 800 900 1000

Temperature (℃)

Str

ain

×1

0E

3

30C/s

50C/s

50℃/s

30℃/s

Strain-Tem

-2

0

2

4

6

8

10

12

100 200 300 400 500 600 700 800 900 1000

Temperature (℃)

Str

ain

×1

0E

3

10C/s

20C/s

25C/s

10℃/s

20℃/s

25℃/s

Page 125: modelling of phase transformation in hot stamping of boron steel

125

Figure 7.3 Tested starting of bainite and martensite transformation temperatures under

cooling rates of 10, 20, 25, 30 and 50℃/s.

The fraction of transformed bainite was measured from the microstructure of heat

treated steel using the image processing program ImageJ. The SEM photographs are

shown in Figure 7.4. The cooling rates of the heat treatment for a), b) and c) were 10℃

/s, 20℃/s and 25℃/s, respectively. Mixed microstructures of bainite and martensite can

be observed. It is clearly shown that the fraction of bainite decreases with the

increasing cooling rate. For each test piece, the fraction of bainite from three locations

have been examined. The results are summarised in Table 7.1, which were used to

validate the bainite transformation model.

a) b)

no deformation

50

200

350

500

650

800

950

1 10 100 1000 10000

Time (s)

Tem

p (

C)

10C/s

20C/s

25C/s

30C/s

50C/s

M

B?

Boron CCT

10℃/s

20

25

30

50

Martensite

Bainite

Ferrite

Pearlite

Page 126: modelling of phase transformation in hot stamping of boron steel

126

c)

Figure 7.4 SEM photographs of the boron steel after the heat treatment test described

in Figure 7.1. Cooling rates of the listed microstructures are 10℃/s, 20℃/s and 25 ℃/s.

Transformed bainite is outlined in the photographs.

Table 7.1 Volume fraction of bainite transformation corresponding to the heat treatment

test in Figure 7.1.

Cooling rate (℃ /s) Average bainite fraction (%)

10 32.3

20 5.31

25 0.78

7.2.2 Bainite and martensite transformation of pre-deformed

material

In order to study the strain effect on martensite and bainite transformation,

deformation was introduced during quenching. The steel was heat treated as shown in

Figure 7.5. The soaking time of 1 minute ensured the steel was thoroughly austenitized

before cooling to the test temperature. The test pieces were tensioned to a strain level

of 0.1 at 1.0/s. Testing temperatures were 800℃ and 600℃. Quenching rates varied

from 15℃/s to 100℃/s. The specimen design is shown in Figure 4.2 with a gauge

length of 30mm.

Page 127: modelling of phase transformation in hot stamping of boron steel

127

Figure 7.5 Test program and temperature profile for phase transformation study when

pre-deformation was introduced.

Following the same procedure as described in the previous section, the start and

end temperatures of bainite and martensite transformation were deduced from

dilatation curves, and the result is summarized in Figure 7.6. The dashed curves are

the CCT diagram of the boron steel. The dots are measured critical temperatures of

bainite and martensite transformation when the steel was pre-deformed at 800℃. The

triangle symbol is measured start temperature of bainite transformation when the steel

was pre-deformed at 600℃. Compared with the CCT diagram of the boron steel in

Figure 7.3, where no pre-deformation was introduced, the start temperature of bainite

transformation increased when the cooling rates are 15℃/s and 25℃/s, the start

temperature of the martensite transformation decreases less than 15℃. The bainite

and martensite transformation temperatures with pre-deformation and without pre-

deformation are listed in Table 7.2.

Time (s)

Isothermal tension: strain=0.1; strain

rate: 1s-1

, T = 800 & 600℃

cooling rate:

15~100℃/s

Cool to testing

temperature at 50℃/s

Heating rate

10℃/s

5℃/s

1.25℃

/s

700

400

925 Soaking time: 1 min

Temp (℃)

Page 128: modelling of phase transformation in hot stamping of boron steel

128

Figure 7.6 Critical temperatures of bainite and martensite transformation when pre-

deformation was introduced.

Table 7.2 Critical phase transformation temperatures in Figure 7.1.

Cooling rate (℃/s) 15 25 30 50 100

Bainite transformation (℃)

(With strain effect) 631 565 - - -

Bainite transformation (℃)

(No strain effect) 592 468 - - -

Martensite transformation (℃)

(With strain effect) 364 - 387 384 385

Martensite transformation (℃)

(No strain effect) 375 - 398 399 400

The microstructure of the steel was examined of that formed at 800℃ and quenched

at 15℃/s (as shown in Figure 7.7 a)), also that formed at 600℃ and quenched at

25℃/s (as shown in Figure 7.7 b)). Mixed phases of bainite and martensite were

obtained in these two cases. Bainite fractions measured after cooling are 38.37% for

the material cooled at 15℃/s, and 22.62% for the material cooled at 25℃/s. For the

same cooling rate, a larger amount of bainite was observed, compared with that in

steel that has not been pre-deformed, as shown in Figure 7.4.

CCT with strain effect

0

100

200

300

400

500

600

700

800

900

1 10 100

Time (s)

Te

mp

era

ture

( ℃

)

25C/s

30C/s

50C/s

Boron CCT

15C/s

100C/s

Tf800&1/s

100℃/s 50℃/s

30℃/s 25℃/s

15℃/s

CCT of boron steel

Page 129: modelling of phase transformation in hot stamping of boron steel

129

a)

b)

Figure 7.7 Microstructures for the material after pre-deformation a) at 800℃ and b) at

600℃, to a strain of 0.1 at 1.0/s. Cooling rates are a) 15℃/s and b) 25 ℃/s. Some

bainite is outlined by dashed circles.

In summary, it is evident from the above results that, when tension to a small strain

(0.1) is introduced during the cooling process:

1) Pre-deformation reduces the martensite transformation temperature by only

approximately 15℃.

2) Bainite transformation temperature increases; nose of bainite transformation

curve moves to the left (reduced time) by introducing pre-deformation.

For pre-deformed test-pieces, a higher quenching rate is needed to obviate

transformation into bainite. The reason for this phenomenon is that pre-deformation

Page 130: modelling of phase transformation in hot stamping of boron steel

130

increases dislocation density inside the steel matrix, which suppresses martensite

transformation. This is due to the fact that the martensite transformation is typically

diffusionless. The mechanism of martensite transformation is by a form of cooperative,

homogeneous movement of many atoms, usually less than the interatomic distances,

that leads to a change in crystal structure, and the atoms maintain their relative

relationships [134]. Large amounts of dislocations may be induced and accumulated at

the grain boundaries during pre-deformation. This interferes with movement of atoms,

changes the carbon diffusion effect on phase transformation and suppresses

martensite transformation. However, the pre-deformation can provide more energy for

bainite transformation which is diffusion controlled. Therefore the start temperature of

bainite transformation is increased.

7.3 Modelling of phase transformation under cooling

processes

The modelling technique of bainite and martensite transformation with and without

pre-deformation is discussed in this section. The mechanism-based unified constitutive

equations, predicting the evolution of bainite during quenching, are proposed. The

critical radius of new formed bainite is deduced, and the evolution rate of the new

formed phase under different cooling rates is evaluated. Optimization procedures are

described to calibrate the unified constitutive equations from heat treatment tests and

relevant microstructure observations.

7.3.1 Modelling of bainite transformation without pre-deformation

7.3.1.1 Nucleation and growth

Theoretically, there is an equilibrium temperature, eqT , between austenite and

bainite phases. The two phases can coexist at this temperature. When the temperature

is higher than eqT , bainite will automatically transform into austenite to maintain the

lowest free energy of the system, and vice versa. In order to achieve the transformation

from austenite to bainite, a small degree of undercooling is required. If the austenite is

undercooled by T ,where eqT T T , before transformation occurs, bainite

transformation will be accompanied by a decrease of free energy VG , which provides

the driving force for the transformation [134], as shown in Figure 7.8. In the figure, eqT

is the equilibrium temperature of austenite and bainite; A

VG and B

VG are free energies

Page 131: modelling of phase transformation in hot stamping of boron steel

131

of initial (austenite) and final (bainite) states respectively. When the undercooling is

T , the corresponding decrease of free energy VG is:

A B

V V V V

eq

TG G G Q

T

7.1

where QV is the activation energy for nucleation of bainite, which is directly proportional

to the driving force for transformation.

Figure 7.8 Changes of volume free energy of bainite/austenite transformation as a

function of temperature [134].

According to the nucleation theory based on homogeneous nucleation, the change

of free energy, when a particle of the new phase is formed, can be ascribed to two

contributions: the decreased free energy per unit volume ( VG ) and the increased

interfacial energy ( ABS ) of small particles. The magnitude of the volume free energy

VG depends on the degree of undercooling. Assuming that a particle of the new

phase is a sphere with radius r, the change of free energy can be written as:

3 24

43

V ABG r G r S 7.2

The interfacial energy ABS is usually difficult to determine, but can be described as

proportional to a temperature ratio [63]:

AB

TS

T

7.3

T eqT

A

VG

B

VG VG

T

r

*r r

Temperature

Free energy

Page 132: modelling of phase transformation in hot stamping of boron steel

132

It is seen from Equation 7.2 that the interfacial energy term increases with 2r ,

whereas the released volume free energy increases with 3r . Therefore the creation of

small particles of solid will always lead to a free energy increase when the radius of the

new formed phase particle is in a certain range. As a consequence, austenite can be

maintained in a metastable state even if the temperature is below Teq. If the radius r is

large, the term 3r will play the key role and the volume free energy term will be

dominant in Equation 7.2. In this case, an increase in value of r will decrease the free

energy of the entire system, as shown in Figure 7.9, and austenite will be decomposed.

Figure 7.9 Change of free energy as a function of nucleus size.

Based on molecular dynamics and kinetics theory, it can be seen from Figure 7.9

that a critical radius, *r , exists which is associated with a maximum excess free energy,

the activation energy barrier *G . Thus,

*r can be determined by differentiating the

change of free energy with respect to radius r , and equating the result to zero, as

follows:

24 8 0V AB

d Gr G rS

dr

7.4

Hereby:

* 2 AB

V

Sr

G

7.5

Substituting Equation 7.1 and 7.3 into Equation 7.5 gives:

24 ABr S

*G

r *r

Interfacial energy

G

Volume free energy

34

3Vr G

0

Page 133: modelling of phase transformation in hot stamping of boron steel

133

1* eq

V

ATr

T TQ

7.6

where A1 is a material constant.

For the nucleated particles of a new phase, only those with radii greater than *r can

grow. The growth rate of the new phase particles is temperature dependent and follows

a Gaussian function, as shown in Figure 7.10. When the temperature is lower than Teq,

a large undercooling will result in a significant decrease of free energy change.

Meanwhile, the carbon concentration in the austenite increases due to the drop in

temperature; hence the carbon diffusion distance is reduced. These factors will assist

the bainite transformation and increase the growth rate. With continuous cooling, the

transformation temperature becomes lower and sharply reduces the activation energy

of phase transformation, which leads to the subsequent decrease of new phase growth

rate. Therefore, a modified Gaussian function is proposed to describe the growth rate

of bainite particle size:

1

1

2

1

exp

nT T

r AB

7.7

where A2, B1, n1 and T1 are material constants. The highest growth rate of r is achieved

at an intermediate temperature T1. When the radius r of the nucleated particle reaches

the critical value r*, the particle can grow larger, and a new phase start arises. The time

for the radius to reach r* is termed the incubation time. Before r reaches r*, the steel is

in an incubation period.

Page 134: modelling of phase transformation in hot stamping of boron steel

134

Figure 7.10 Growth rate of the size of bainite as a function of temperature.

7.3.1.2 Evolution of volume fraction

The definition of the start of formation of a new phase is when the radius of the new

phase particle r is greater than *r . The fraction of new formed phase increases with

increasing size of the new phase particle. Therefore, the term

*r r

r

is introduced to

take into account the effect of the radius of new phase particle, and normalizes the

term into the range 0 to 1. During the incubation period, if *r r , the growth rate of

volume fraction 0X .

When bainite begins to form, the nucleation rate, which is temperature dependent,

becomes one of the major factors dominating the amount of transformed phase. The

relation between nucleation rate and temperature can be considered as a modified

Gaussian distribution as shown in Figure 7.11. When the undercooling is less than T ,

nucleation is negligible as the driving force is too small for a nuclei to grow. The

nucleation rate increases with the increasing undercooling in a temperature range

when 2T T . With very large undercooling, when 2T T , the nucleation rate will

decrease because of slow diffusion due to the low temperature. A maximum nucleation

rate is obtained at an intermediate temperature, T2. The effect of temperature on the

amount of phase transformation can be introduced by the nucleation rate, in a form of a

modified Gaussian function.

Growth rate r

Temperature T

Teq

T1

Page 135: modelling of phase transformation in hot stamping of boron steel

135

Figure 7.11 Nucleation rate of bainite as a function of temperature.

Considering the effects of new phase particle size, and nucleation rate, the growth

rate of bainite fraction X for the transformation from austenite to bainite can be

presented as

32

4

*2

3

2

exp 1

nn

nT Tr rX A r X

r B

7.8

where A3, B2 and T2 are material constants, the powers n2, n3, n4 give higher regulation

of the equation. The term 41n

X is introduced to ensure that the volume fraction of

phase transformation X varies from 0.0 to 1.0, as well as to control the effect of the

transformed phase on the transforming phase because the growth rate X will

decrease with the accumulation of transformed phase. For the term

2*n

r r

r

, only

positive values are considered: if * 0r r , 0X . Since only two phases, bainite and

martensite, exist in the hot stamped parts in the investigated range of cooling, the

fraction of martensite XM can thus be obtained by:-+

1m BX X 7.9

7.3.1.3 Unified constitutive equations

By introducing the critical radius of the new phase particle, the grain size growth rate

and the volume fraction growth rate of bainite, a set of unified constitutive equations

Nucleation rate

Temperature

Teq

T2

T

Page 136: modelling of phase transformation in hot stamping of boron steel

136

predicting the volume fraction of bainite transformation during the quenching of boron

steel are formulated as:

1

32

4

1*

1

2

1

*2

3

2

exp

exp 1

eq

eq V

n

nn

n

ATr

T T TQ

T Tr A

B

T Tr rX A r X

r B

7.10

where A1, A2, A3, T1, T2, B1, B2, n1, n2, n3, n4 and QV are material constants, and need to

be determined from experimental results and CCT diagram. Temperature is in Kelvin.

Teq is the equilibrium temperature which is given in the CCT diagram. For boron steel,

Teq is 983 K when no pre-deformation is introduced.

7.3.1.4 Calibration of the model

In Equation set 7.10, the equations of *r and r predict the start of bainite

transformation as the phase transformation starts when *r r . Thus, the material

constant in equations of *r and r , which are T1, n1, B1, A1, A2, QV, ands Teq, can be

determined by fitting the start temperature and time of bainite transformation with the

CCT diagram.

Comparison of the best fitting of computed starting temperature of bainite

transformation and CCT diagram of boron steel is shown in Figure 7.12. Cooling rates

vary from 10℃/s to 25℃/s, which are in the range of cooling rates when a mixed phase

of bainite and martensite can be obtained. The triangle symbols are the computed

temperatures of bainite transformation under cooling rates of 10, 15, 20 and 25℃/s. It

is seen that the predicted result conforms well with the CCT diagram. Determined

constants in equations of *r and r are listed in Table 7.3.

Page 137: modelling of phase transformation in hot stamping of boron steel

137

Figure 7.12 Comparison of predicted starting temperature of bainite transformation and

CCT diagram of boron steel.

Table 7.3 Constants determined by fitting the predicted starting temperature of bainite

transformation with CCT diagram.

T1 (K) n1 (-) B1 (-) A1 (µm/s) A2 (µm/s) QV (J/mol)

980 3.542 7.408e7 6.193e6 0.7057 124500

To determine the constants in the equation of X , the model was fitted to the

fraction of transformed bainite. Microstructures of heat treated material have been

examined, and results were presented in Table 7.1. The SEM photographs are shown

in Figure 7.3. Cooling rates are 10℃/s, 20℃/s and 25℃/s, respectively. Comparison of

predicted bainite fraction and experimentally observed bainite fraction are shown in

Figure 7.13. A good agreement has been achieved and the constants determined are

listed in Table 7.4.

CCT of boron steel

0

200

400

600

800

1000

1 10 100 1000 10000

Time (s)

Te

mp

era

ture

(℃)

F P

M

B Predicted

10℃/s 15℃/s 20℃/s

25℃/s

Page 138: modelling of phase transformation in hot stamping of boron steel

138

Figure 7.13 Comparison of predicted bainite fraction and experimentally observed

bainite fraction.

Table 7.4 Constants determined by fitting the predicted fraction of bainite

transformation with experimentally measured fraction of bainite transformation.

A3 (s-1) B2 (-) n2 (-) n3 (-) n4 (-) T2 (K)

0.0886 100.396 0.009 80.551 1.45 813

7.3.2 Modelling of bainite transformation of pre-deformed material

Pre-deformation has been reported to affect the start temperature, as well as the

amount of bainite transformation [128-131], in a complex manner and thus is of

importance when forming high strength steels [198]. The plastic deformation introduced

before bainite transformation changes the energy that is required to drive the phase

transformations which was found in the tests used to promote bainite transformation. It

is therefore necessary for this to be quantified. In this section, an energy-based

material model which can be used to formulate the start temperature and to predict the

fraction of bainite transformation as a function of pre-strain is established.

7.3.2.1 Strain effect

The effect of strain on the bainite transformation was modelled by Garrett by

introducing an strain energy density factor [63]. However, the strain energy density

factor can not cover all the effect of strain on the bainite transformation in practice; for

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 5 10 15 20 25 30 35 40

Time (s)

Bain

ite fra

ctio

n

32.30%

5.31%

0.78%

Measured bainite fraction after heat treatment

Computed evolution of

fraction rate X ,10℃/s

20℃/s

25℃/s

Page 139: modelling of phase transformation in hot stamping of boron steel

139

example during the relaxation process after hot rolling of steels, as discussed in

Chapter 3. Therefore a new model is introduced in which the strain effect is accounted

for by introducing the energy factor f into Equation 7.1.

A B

V V V V

eq

TG G G Q f

T

7.11

Since the promotion of the bainite transformation by pre-deformation is believed to

be due to the accumulation of dislocation, the energy factor is thus defined as

Bf 7.12

Consequently

A B

V V V V B

eq

TG G G Q

T

7.13

where B is a material constant having the same unit as f . The unit is J/mol.

Therefore

1* 2 eqAB

eqVV B

ATSr

TTGT TQ

T

7.14

By this approach, the strain effect on bainite transformation is represented by

dislocation density instead of external force or strain. For example, during relaxation,

although the external force is removed, the dislocations that exist gradually decreases

to zero if the time is sufficient long. Thus the strain effect on the bainite transformation

decreases with decreasing dislocation density according to the model, which accurately

describes the real situation. Strain rate effect on bainite transformation can also be

described by including the term f into the equation for VG . A higher strain rate

results in a higher dislocation density, consequently, a higher volume fraction of bainite

in the final phase; a lower strain rate allows for longer time of relaxation which results in

a lower dislocation density, therefore, a small rate of deformation has less pronounced

promotion of the bainite transformation [199, 200].

Equations of r and X keep the same form as shown in Equation 7.10. The new

equation set is shown in Equation 7.15. The constants need to be re-determined by

fitting with the experimental data as shown in Figure 7.6. The compared result is shown

in Figure 7.14. The solid curves in Figure 7.14 a) and b) are predicted CCT diagram

Page 140: modelling of phase transformation in hot stamping of boron steel

140

when the steel is pre-deformed at 800℃ and 600℃ to a strain of 0.1 at 1.0/s, and the

dots in Figure 7.14 a) and b) are measured results. The thick dashed curve in Figure

7.14 a) is the predicted CCT diagram when the steel is pre-deformed at 800℃ to a

strain of 0.05 at 1.0/s.

1

32

4

1*

1

2

1

*2

3

2

exp

exp 1

eq

eq

V eq B

eq

n

nn

n

ATr

TTTQ T T

T T

T Tr A

B

T Tr rX A r X

r B

7.15

a) b)

Figure 7.14 Comparison of experimentally obtained CCT diagram of boron steel with

strain effect (dots) and the predicted data (solid curves and the thick dashed curve) by

the proposed model. a) pre-deformation is at 800℃, b) pre-deformation is at 600℃.

The determined constants in equations for *r and r are listed in Table 7.5. It is

shown from Figure 7.14 that the predicted start temperatures of bainite transformation

agree well with the experimental observed values, when pre-deformation is introduced.

CCT with strain effect

0

100

200

300

400

500

600

700

800

900

1 10 100

Time (s)

Te

mp

era

ture

( ℃

)

25C/s

30C/s

50C/s

Boron CCT

15C/s

100C/s

Tf800&1/s

800C-0.1-1/sno strain

800C-0.05-1/sno strain MsEXPno strain BsEXP

100℃/s

50℃/s

30℃/s 25℃/s

15℃/s

Experimental

Predicted

pre-formed at 1/s to 0.1 at 600℃,

0

100

200

300

400

500

600

700

800

900

1 10 100

Time (s)

Te

mp

era

ture

(℃

)

25C/s

30C/s

50C/s

Boron CCT

15C/s

100C/s

Tf600str0.1&25C/s&1/sno strainpredictedno strain

600C, 1/s,0.1

100℃/s

50℃/s

30℃/s 25℃/s

15℃/s

Experimental

Predicted

Page 141: modelling of phase transformation in hot stamping of boron steel

141

Table 7.5 Determined material constants for equations of *r and r .

T1 (K) n1 (-) B1 (-) A2(µm/s) A1(µm/s) QV(J/mol) Teq (K) B (J/mol)

980.0 3.542 7.408e7 0.7057 6.893e6 210550.0 983.0 1290.0

In order to determine the constants in the equation for X , the fractions of bainite

were measured in SEM photographs for the steel that had experienced pre-

deformation at 800℃ (Figure 7.7 a)) and 600℃(Figure 7.7 b)), to a strain of 0.1 at

1.0/s. In both figures, a mixed microstructure of martensite and bainite was observed.

By comparing the experimentally measured bainite fraction with the predicted value, as

summarised in Figure 7.15, the constants were determined and listed in Table 7.6. In

Figure 7.15, the solid curves are the predicted evolution of bainite fraction and the

outline circles are experimentally measured values. The pre-deformation temperatures

are 600℃ and 800℃, respectively. Both test-pieces were tensioned to a strain of 0.1,

at 1.0/s. Fraction of bainite increased with increasing time and decreasing temperature,

until the temperature drops below Ms. Martensite transformation is almost

instantaneous, which occurs as soon as the temperature reaches Ms. Good agreement

between the predicted and experimental results has been achieved. The proposed

bainite transformation model can successfully predict the evolution of bainite

transformation in the investigated boron steel.

Figure 7.15 Comparison of predicted bainite fraction and experimentally observed

bainite fraction, with strain effects. The temperatures of pre-deforming are 600℃ and

800℃, respectively. The strain rate is 1/s, strain is 0.1.

Vb-time

0

10

20

30

40

50

10 15 20 25 30 35

Time (s)-cooling from 900℃

Ba

init fra

ctio

n %

Expermental

800C-15C/s-0.1-1/s

600C-25C/s-0.1-1/s

strain=0

pre-deformed at 600℃

Predicted curves,

pre-deformed at 800℃

Experimental value,

pre-deformed at 800℃

Without pre-deformation

Page 142: modelling of phase transformation in hot stamping of boron steel

142

Table 7.6 Determined constants for the equation for X .

A3 (s-1) B2 (-) n2 (-) n3 (-) n4 (-) T2 (K)

0.27 80.396 0.009 80.551 1.45 813

7.4 Validation of the material models

FE analysis of the formability tests using a commercial software was conducted. The

FE analysis results were compared with those of experiments. Good agreement has

been achieved.

7.4.1 Implementation of the material models

Constitutive material models can be implemented into ABAQUS/Explicit using

VUMAT. The VUMAT subroutine is called for blocks of material calculation points

during the FE analysis. It can use and update solution-dependent state variables, e.g.

plastics strain (p

e ), dislocation ( ), damage ( D ) and the amount of transformed

bainite ( X ). The developed viscoplastic damage model and phase transformation

model were programmed into VUMAT using the FORTRAN language.

7.4.2 Simulation results

The FE model of formability test built by ABAQUS is shown in Figure 7.16 a). Since

the blank holders and punch are all axisymmetric, a 2-D axisymmetric model was used

to save the computational cost. The initial temperature of blank holders and punch is

20℃. The initial temperature of steel sheet is 700℃. The upper blank holder moves

down at constant speeds: 15 mm/s and 100 mm/s. An evenly distributed blank holder

force is applied to the lower blank holder, after the initial contact. The initial force is

1.13kN. The force linearly increased with the displacement of upper die until it reached

the maximum blank holder force, 1.76 KN, which simulates the supporting force from

the gas springs of the dies. The gas spring constant is 10N/mm. In the FE model, the

mesh of the blank is quad structured, the mesh of the punch and blank holders are

quad-diamond free. The elements are coupled temperature-displacement type. Heat

transfer coefficient with functions of pressure and clearance are given in Figure 7.17.

Figure 7.16 b) and c) are the modelled formed test pieces that were stamped at ram

speeds of 100 mm/s and 15 mm/s, respectively.

Page 143: modelling of phase transformation in hot stamping of boron steel

143

a)

b)

c)

Figure7.16 a) FE model of formability test built by ABAQUS. b) FE analysis of formed

test piece when the ram speed is b) 100 mm/s and c) 15 mm/s.

Damage

Damage

40mm

27mm

R=3mm R=40mm

50 m

m

Page 144: modelling of phase transformation in hot stamping of boron steel

144

Figure 7.17 Heat transfer coefficient with functions of pressure and clearance.

The profiles of the half-cross-section of the formed test pieces were compared with

FE analysis result, as shown in Figure 7.18. Ram speeds of a) and b) are 100 mm/s

and 15 mm/s, respectively. The predicted and experimentally measured thickness

distributions are shown in Figure 7.19. Evolution of the central hole diameters is shown

in Figure 7.20. Good agreement has been achieved between the experimental results

and FE analysis results. The implemented viscoplastic damage model can predict the

failure mode during hot stamping of the boron steel accurately.

a) b)

Figure 7.18 Comparison of half-cross-section profiles between the experimental result

and FE analysis. Ram speeds are a) 100 mm/s and b)15 mm/s.

Predicted

Experimental

Predicted

Experimental

Heat transfer coefficient-clearance

0

0.5

1

1.5

2

2.5

01234

clearance (mm)

he

at tr

an

sfe

r co

effic

ien

t

(N/s

ec/m

m/K

)

Heat transfer coefficient-pressure

0

0.5

1

1.5

2

2.5

0 3 6 9 12

pressure (MPa)h

ea

t tr

an

sfe

r co

effic

ien

t

(N/s

ec/m

m/K

)

0

0

0.5

1

1.5

2

2.5

mW/(mm2K)

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145

Figure 7.19 The predicted and experimentally measured thickness distribution.

Figure 7.20 Comparison of the evolution of central hole diameters between the

experimental result and FE analysis. Ram speeds are 100 mm/s and 15 mm/s.

The predicted bainite distribution in the cross-section area obtained from the FE

analysis was compared with the microstructure observations on the test specimens.

Results are shown in Figure 7.21. It is seen from Figure 7.21 a) that when the ram

speed is 100mm/s, more bainite is obtained in the area close to the central hole of the

formed test piece. This is due to the high dislocation density resulting from the large

deformation in this area, which promotes bainite transformation. The left tip of the test

piece shown in the figure (the central hole edge) has a relative low amount of bainite.

This is because the central hole edge contacted the cold punch first during stamping;

Thickness distribution

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

X-Position (mm)

No

rma

lise

d t

hic

kne

ss t

/t0

(m

m)

15FE

100FE

15ave

100average

100 mm/s; exp

100 mm/s; comp

15 mm/s; exp

15 mm/s; comp

dt/d0 vs. st

1

1.2

1.4

1.6

1.8

0 5 10 15 20

Displacement (mm)

dt/

d0

15 mm/s

100 mm/s

15FE

100FE15mm/s; comp

15mm/s; exp

100mm/s; comp

100mm/s; exp

Page 146: modelling of phase transformation in hot stamping of boron steel

146

therefore heat transfer is most significant in this area, the highest cooling rate of the

test piece is achieved in this area. Consequently, more martensite is transformed and

less bainite is observed, although the dislocation density is high. In Figure 7.21 b),

highest fraction of bainite transformation (B%=1-M%) is observed at the failure area

which also has the highest dislocation density due to the large deformation.

a)

b)

Figure 7.21 Comparison of predicted martensite distribution and observed bainite

distribution. Ram speeds of tests are a) 100 mm/s and b) 15 mm/s. Some bainite is

highlighted by dashed circles.

In this chapter, a phase transformation model which can describe the evolution of

fraction of bainite transformation was introduced. The effect of strain on a bainite

Martensite%

Martensite%

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147

transformation is taken into account. The model was validated by the formability tests.

Good agreement was obtained between the FE analysis and the SEM photographs.

The developed viscoplastic damage model and bainite transformation model can

describe the formability of boron steel and the evolution of bainite transformation

accurately.

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148

Chapter 8

Conclusions and suggestions for

future research

8.1 Conclusions

A knowledge of phase transformations in the process is critical for determining

physical and mechanical properties of formed parts. The research work described in

this thesis deals with the modelling of phase transformation in hot stamping and cold

die quenching of boron steel.

1) Austenitization:

Austenitization of steel is a typical carbon diffusion process, which can be described

by nucleation, grain growth and homogenization processes. It is experimentally

observed that the evolution of austenite is multi-controlled by: diffusion coefficient,

temperature, heating rate and the accumulation of austenite. In a continuous heating

process, a higher heating rate results in a longer incubation period and a higher

austenitization start temperature. Heating rate has a greater effect on the completion

temperature than on the start temperature of austenitization. Austenitization takes

place after incubation and the phase transformation is fast at the beginning. With the

accumulation of austenite, the austenitization process slows down. By comparing the

dilatation curves from the heat treatment tests at different heating rates, it is seen that a

slower heating rate results in a greater proportion of austenite. The reason is that,

austenitization is diffusion controlled, thus is significantly affected by time and

temperature. For a particular targeted temperature, a lower heating rate allows for a

longer time for austenite transformation.

A set of unified constitutive equations were proposed to model the austenitization of

boron steel in continuous heating processes, e.g. describing the relationship between

time, temperature and an austenitization process. In the model, a variable ranging from

0, (corresponding to the beginning of the incubation period) to 1, (corresponding to the

end of the incubation period) is introduced to describe the incubation period. The

evolution rate of the volume fraction of austenite is proportional to the diffusion

coefficient, and multi-affected by heating rate and the fraction of austenite that has

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149

already transformed. The evolution of dilatation strain was deduced from the dilatation

tests, which is a function of thermal expansion coefficient, heating rate, lattice constant,

and volume fraction of austenite. The austenitization model was calibrated by fitting

outputs from it with dilation curves obtained from heat treatment tests. A good

agreement has been obtained between experimental and computed data. Dilatometry

tests are effective in the study of phase transformation of steels.

2) Bainite and martensite transformation:

Due to the high cooling rates during cold die quenching, only bainite and martensite

transformations were observed in the cold die quenching of boron steel. It was shown

from the dilatometry tests that the fraction of bainite decreases with increasing cooling

rate. Pre-deformation affects the bainite and martensite transformations. By introducing

a pre-deformation strain of 0.1, the martensite transformation temperature Ms was

decreased by approximately 15℃. Pre-deformation promotes bainite transformation. In

the CCT diagram for boron steel, the nose of bainite transformation moves left; at a

certain cooling rate, bainite fraction increased, and less volume fraction of martensite

was obtained in the final microstructure of the formed test-pieces. It is believed that the

pre-deformation in the austenite state increases dislocation density. The higher

dislocation density provides more driving force for bainite transformation, and facilitates

the nucleation and grain growth of bainite.

A bainite transformation model in the form of unified constitutive equations was

proposed to describe the effect of strain on the evolution of volume fraction of bainite

transformation. In the model, modified Gaussian functions are used to describe the

evolution of growth rate of bainite particle size, and nucleation rate of bainite. The

critical radius of bainite particles is deduced under the condition when the free energy

of bainite nucleation and growth reaches a maximum value: the activation energy

barrier. Pre-deformation provides more driving force for bainite transformation by a

magnitude of f which is a function of normalised dislocation density: Bf . The

model can thus predict both strain and strain rate effects on the bainite transformation.

A higher strain rate results in a higher dislocation density, consequently, a higher

volume fraction of bainite in the final phase. A lower strain rate allows for a longer

relaxation time which results in lower dislocation density, therefore, less bainite is

obtained in the final microstructure. The model was calibrated by fitting outputs from it

with dilatation curves obtained from dilatometry tests, when pre-deformation was

introduced. A good agreement was obtained in a comparison of the experimental

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150

curves with the predicted curves. The fraction of martensite, M%, is calculated by

M%=1-B%, where B% is the volume fraction of bainite.

3) Thermal mechanical properties and damage mechanism:

Mechanical properties of boron steel were determined using a Gleeble thermal

simulator. The temperature distribution in heated test-pieces of different gauge lengths

was measured. When the gauge length was shorter (e.g. 15 mm), the temperature

gradient was much larger in the central area; hence tensile deformation is likely to be

localized at the centre. Consequently, a large gauge length (e.g. 50mm) was employed

for isothermal tension to minimise the effect of localised deformation. Uni-axial tensile

tests were conducted to study the mechanical properties: boron steel exhibits obvious

viscoplastic behaviour when forming in the temperature range from 600℃ to 800℃.

The maximum flow stress increases with increasing strain rate.

A viscoplastic-damage model in the form of constitutive equations was developed to

describe the mechanical behaviour of boron steel. The concept of normalised

dislocation density is employed to describe working hardening of the material. The

evolution rate of the normalised dislocation density is multi controlled by plastic strain

rate, dynamic recovery, and static recovery. A damage factor D, which determines the

level of damage and strain to failure is proposed. The evolution rate of damage is

proportional to the material flow stress and the plastic strain rate p , and is a

function of temperature. The factor 2

1

1f D

D

is employed to rationalise the

significant damage growth at large strain states. The parameters in the model were

determined by fitting with the experimental stress-strain curves from uni-axial tensile

tests.

4) Validation and Finite Element Analysis:

Formability tests were conducted to reveal the deformation behaviour and phase

transformation of the steel in hot stamping and cold die quenching and validate the

material models. From the formability tests, circumferential and radial crack failure

modes were observed at a slow forming speed, 15mm/s, and high forming speed, 100

mm/s, respectively. This was believed to be due to an uneven temperature distribution,

resulting in an uneven hardness distribution in the test-pieces, during stamping.

Deformation was evaluated by comparing the central hole diameters of the test-pieces

performed at different forming speeds. Only bainite and martensite were observed in

the formed test-pieces.

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151

FE analysis was carried out to simulate the formability tests using the commercial

software, ABAQUS. The developed viscoplastic-damage model and bainite

transformation model were integrated with ABAQUS using a user defined subroutine

VUMAT. A good agreement was obtained between the experimental and FE results for:

deformation degree, thickness distribution, and microstructure evolution. The

viscoplastic-damage model and bainite transformation model accurately predict the

viscoplastic behaviour and bainite and martensite transformation of boron steel in the

hot stamping and cold die quenching process.

8.2 Suggested future work

The constitutive models developed in this thesis for hot stamping and cold die

quenching process show great promise for future work. For example, bainite

transformation is modelled in this thesis and the predicted results agree well with the

experimental results. This is sufficient to describe phase transformation of boron steel

in a cold die quenching process; however, other phase transformations may occur in

slow cooling situations and these should be modelled. Also the effect of strain on other

phase transformations, for example, ferrite or pearlite, should be determined.

Furthermore, knowledge of the effect of strain at large strain levels, e.g. greater than

0.1, is important to describe phase transformation after large deformation.

To maximise advantages of the hot stamping and cold die quenching process and

enable further weight reduction of formed parts, it is desirable to produce panel parts

having particular mechanical properties in different regions [201-204]. This could be

feasible by introducing different controlled cooling rates in different regions of a work-

piece. Thus a tailored microstructure distribution within one component can be

obtained. This concept has great potential in automotive applications for producing

advanced safety critical components. The technique to enable panel parts to be formed

with a tailored microstructure distribution (and hence a tailored mechanical property

distribution) is of great interest for passenger car manufacturers and steel companies.

Constitutive models which can predict various microstructure distributions, as well as

the localised mechanical properties due to the microstructure distribution is meaningful

in improving the conventional hot stamping and cold die quenching process.

Constitutive material models based on physical mechanisms will continue to be an

important method of describing complex material behaviour for the foreseeable future.

This thesis enhanced the understanding of phase transformation of boron steel in both

heating and cooling situations and the viscoplastic behaviour of boron steel at different

forming temperatures and strain rates. The methodology and results are transferable to

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152

other forming techniques and materials. Research work can be extended to other types

of boron steels, as well as other types of quenchable steels. However, this research

area is full of challenges, e.g. development of physical mechanism based models,

determination and validation of models with high precision. The material constants

were determined from the best fittings between the experimental data and computed

data, thus the determination of these constants greatly depends on the test results.

Sensitivity study of these parameters will help to improve the material models.

Page 153: modelling of phase transformation in hot stamping of boron steel

153

References

1. Bloomfield, G.T., The world automotive industry. 1978, Newton Abbot; London;

North Pomfret (Vt); Vanvouver: David and Charles.

2. Williams, B., Evolution of Modern Automobile Industry.

http://ezinearticles.com/?Evolution-of-Modern-Automobile-Industry&id=2094366,

2009.

3. Climate Change 2001: The Scientific Basis-Summary for Policymakers: A

Report of Working Group 1 of the Intergovernmental Panel on Climate Change.

2001, Intergovernmental Panel on Climate Change (IPCC): Geneva,

Switzerland.

4. 2008; Available from: Automobile Industry:

http://www.history.com/encyclopedia.do?articleId=201868.

5. The Kyoto Protocol to the United Nations Framework Convention on Climate

Change. Available from: http://unfccc.int/resource/docs/convkp/kpeng.pdf.

6. Austin, D., et al., Changing drivers : the impact of climate change on

competitiveness and value creating in the automotive industry. 2003, Zurich:

Sustainable Asset Management, World Resources Institute. viii, 73 p.

7. Jeanneau, M. and P. Pichant, The trends of Steel Products in the European

Automotive Industry. La Revue De Metallurgies-CIT, 2000: p. 1399-1408.

8. Oren, E.C. Automotive Materials and Technologies for the 21st Century. in 39th

MWSP 1998. Indianapolis, USA.

9. World Report on Road Traffic Injury Prevention. 2004, World Health

Organization: Geneva.

10. Jacobs, G., A. Aeron-Thomas, and A. Astrop, Estimating Global Road Fatalities.

2000, Transport Research Laboratory.

11. Broughton, J., The Numerical Context for Setting National Casualty Reductions

Targets. 2000, Transport Research Laboratory Ltd: Crowthorne.

12. Jeswiet, J., et al., Metal forming progress since 2000. CIRP Journal of

Manufacturing Science and Technology, 2008. 1(1): p. 2-17.

13. Kleiner, M., M. Geiger, and A. Klaus, Manufacturing of Lightweight Components

by Metal Forming. CIRP Annals - Manufacturing Technology, 2003. 52(2): p.

521-542.

14. Mayer, H., F. Venier, and K. Koglin, Die ASF-Karosserie des Audi A8. The new

Audi A8-special edition of ATZ/MTZ, 2002: p. 94-108.

15. 2009; Available from:

http://www.alcoa.com/aats/en/product.asp?cat_id=1527&prod_id=2956.

Page 154: modelling of phase transformation in hot stamping of boron steel

154

16. Zengen, K.-H.v. Aluminium-the Light Body Material. in New Advances in Body

Engineering. 2002. ika Aachen.

17. Langerak, N. and S. Kragtwijk. The Application of Steel and Aluminum in a New

Lightweight Car Body Design. in Proceeding International Body Engineering

Conference (IBEC) 1998. Detroit MI, USA: Society of Automotive Engineerings

(SAE).

18. Schaik, M.v. and B. Kagi, SOUTUBE, a New Way of Tube Manufacturing. Tube

International 1999. 18(96): p. 181-183.

19. Neugebauer, R., Application of the Parallel Kinematic Machine Principle in a

New Hydraulic Powered, Flexible Bending Machine for Tubes and Profiesl, in

3rd Chemnitz Parallel Kinematics Seminar. 2002: Chemnitz. p. 629-638.

20. Arnet, H., Section Bending with Kinematic Shaping. Advanced Technology of

Plasticity, 1999. 3: p. 2349-2354.

21. Gerger, M. and A. Sprenger, Controlled Bending of Aluminium Extrusions.

Annals of the CIRP, 1998. 47(1): p. 197-202.

22. Nock, M. and M. Geiger, Flexible Kinematic 3D-bending of Tubes and Profiles.

Advanced Technology of Plasticity, 2002. 1: p. 643-648.

23. Kleiner, M., Method and Device for the Production of Curved Workpieces, in

European Patent Specification. 1999.

24. Gusinde, A. and M. Hoogen. Aluminiumblechwerkstoffe, Ensatz in der

Automobilhersteellung. in Landshuter Leichtbaukolloquium. 2003. Landshut.

25. Pirchner, H., M. Weber, and R. Kawalla. Magnesiumbleche für den

Karosseriebau. in Neuere Entwicklungen in der Blechumformung, MAT INFO.

2002. Frankfurt.

26. Flegel, A. Geschlossene Prozessketten vom Halbzeug zum Bauteil aus der

Sicht des Automobilherstellers. in Neuere Entwichlungen in der

Blechumformung, MAT INFO. 2002. Frankfurt.

27. Langerak, N. and S. Kragtwijk, Lightweight Car Body Design. Automotive

Engineering International, 1998. 10: p. 106-110.

28. Schmoeckel, D., et al., Metal Forming of Tubes and Sheets with Liquid and

Otehr Flexible Media. Annals of the CIRP, 1999. 48(2): p. 497-513.

29. Siegert, K. and M. Aust, Hydromechanical Deep-Drawing. Production

Engineering, 2000. 7(2): p. 7-12.

30. Kleiner, M., W. Homberg, and A. Brosium, Process and Control of Sheet Metal

Hydroforming. Advanced Technology of Plasticity, 1999. 2: p. 1243-1252.

31. Homberg, W., Untersuchungen zur Prozessführung und zum Fertigungssystem

bei der Hochdruck-Blech-Umformung. 2000, Dortmund University.

32. WorldAutoSteel. [cited 2009; Available from: www.worldautosteel.org.

Page 155: modelling of phase transformation in hot stamping of boron steel

155

33. UltraLight SteelAutoBody-Advanced Vehicle Concepts (ULSAB-AVC) Overview

Report. 2002, International Iron and Steel Institute: www.worldautosteel.org.

34. Hein, P., A Global Approach of the Finite Element Simulation of Hot Stamping.

Advanced Mat.Res, 2005. 6(8): p. 763-770.

35. Advanced High Strength Steel (AHSS) Application Guidelines, Version 4.1.

2009: www.worldautosteel.org.

36. Miller, W.S., et al., Recent development in aluminium alloys for the automotive

industry. Materials Science and Engineering A, 2000. 280(1): p. 37-49.

37. Schumann, S. and H. Friedrich. Current and Future Use of Magnesium in the

Automobile Industry. in Proceedings of Magnesium Alloys. 2003. Osaka, Japan.

38. Friedrich, H. and S. Schumann. The Second Age of Magnesium Research

Strategies to Bring the Automotive Industry's Vision to Reality. in Magnesium

2000. 2000. Dead Sea, Isreal.

39. Ford Motor Company News Release, March. 2008.

40. WorldsteelAssociation. Advanced steel applications help to address climate

change. 2008.

41. Mercedes-Benz. http://www2.mercedes-benz.co.uk/.

42. Karbasian, A. and A.E. Tekkaya, A Review on Hot Stamping. Journal of

Materials Processing Technology, 2010. 210: p. 2013-2118.

43. Hein, P. and J. Wilsius, Status and innovation trends in hot stamping of

USIBOR 1500 P. Steel research international, 2008. 79(2): p. 85-91.

44. Fujita, T., T. Urabe, and M. Sakurai, High-Performance, High-Strength Steel

Sheets for Exposed Auto Body Panels. JFE GIHO, No.16, 2007: p. 12-15.

45. Cai, J., J. Lin, and T. Dean, A Novel Process: Hot Stamping and Cold Die

Quenching, in 3rd International Symposium on Advanced Technology for

Plasticity. 2007: Nanchang.

46. Kleiner, M., S. Chatti, and A. Klaus, Metal forming techniques for lightweight

construction. Journal of Materials Processing Technology, 2006. 177(1-3): p. 2-

7.

47. Carlsson, B., Formability of High Strength Dual Phase Steels, in SSAB Tunnplåt

AB. 2004: Borlänge, Sweden.

48. Takahashi, M., Development of High Strength Steels for Automobiles. Nippon

Steel Technical Report, No 88, 2003.

49. Chatti, S., et al., Forming and Further Processing of Tailor Rolled Blanks for

Lightweight Structures. Advanced Technology of Plasticity, 2002. 2: p. 1387-

1392.

Page 156: modelling of phase transformation in hot stamping of boron steel

156

50. Wilsius, J., P. Hein, and R. Kefferstein, Status and Future Trends of Hot

Stamping of USIBOR 1500P®, in Tagungsband zum1, in Erlanger Workshop

Warmblechunformung. 2006.

51. Hein, P. and M. Geiger, Advanced Process Control Strategies for the

Hydroforming of Sheet Metal Pairs. Advanced Technology of Plasticity, 1999. 2:

p. 1267-1272.

52. Kleiner, M., A. Gartzke, and R. Kolleck, Experimental and Finite Element

Analysis of Capabilities and Limits of A Combined Pneumatic and Mechanical

Deep Drawing Process. Annals of the CIRP 1997. 46(1): p. 201.

53. Siegert, K. and M. Aust, Hydromechanical Deep-drawing, Production

Engineering. Annals of the German Academic Society for Production

Engineering, 2000. 7(2): p. 7-12.

54. Friebe, E., Hydromechnical Deep Drawing of Passenger Car Fuel Tanks, in

Hydroforming of Tubes, Extrusions and Sheets. 2001: Frankfurt. p. 193-213.

55. Kleiner, M., W. Homberg, and A. Wellendorf. Sheet Metal Hydroforming for Car

Body Applicaitons. in Proceedings of CBC. 2002. Chemnitz, Germany.

56. Yoshitake, A. and K. Yasuda, Vision of Application Technologies for High

Strength Steel Sheets supporting Automabile Weight Reduction. JFE GIHO

2007. No.16: p. 6-11.

57. Merklein, M. and M. Geiger, New materials and production technologies for

innovative lightweight constructions. Journal of Materials Processing

Technology, 2002. 125-126: p. 532-536.

58. Flehmig, T., K.W. Blumel, and M. Kibben, Thin-Walled Steel Tube Pre-Bending

for Hydroformed Components-Bending Boundaries and Presentation of a New

Mandrel Design, in SAE International. 2001: Detroit, USA.

59. Gerlach, J., et al., Material Aspects of Tube-Hydroforming, in SAE International.

1999: Detroit, USA.

60. Li, G.Y., M.J. Tan, and K.M. Liew, Three-dimensional modeling and simulation

of superplastic forming. Journal of Materials Processing Technology, 2004.

150(1-2): p. 76-83.

61. Zhang, Z. and R.A. Farrar, An Atlas of Continuous Cooling Transformation (Cct)

Diagrams Applicable to Low Carbon Low Alloy Weld Metals. 1995: Maney

Publishing.

62. Duque Múnera, D., F. Pinard, and L. Lacassin. Very an Ultra High Strength

Steels Based Tailored Welded Blanks: A Step Further Towards

Crashworthiness Improvement. in SAE 2006 World Congress. 2006. Detroit, MI,

USA.

Page 157: modelling of phase transformation in hot stamping of boron steel

157

63. Garrett, R.P., et al., A Model for Predicting Austenite to Bainite Phase

Transformation in Producing Dual Phase Steels. International Journal of

Machine Tools and Manufacture, 2004. 44: p. 831-937.

64. Shulkosky, R.A., et al., A Microstructure Evolution Model Used for Hot Strip

Rolling, in Material Science & Technology Conference 2003. 2003: Chicago,

Illinois.

65. Cai, J., et al., Austenitization mechanisms and modelling methods for steels.

World Journal of Engineering, 2008. 5(1): p. 10-20.

66. Brooks, C.R., Principles of the austenitization of steels. 1992: Elsevier Applied

Science.

67. Thelning, K.-E., Steel and its heat treatment. 2ed ed. 1984: Butterworth & Co.

68. Speich, G.R. and S. A., Formation of Austenite from Ferrite and Ferrite-Carbide

Aggregates. Transactions of The Metallurgical Society of the American Institute

of Mining,Metallurgical and Petroleum Engineers, 1969. 245: p. 1063-1074.

69. Grossman, M.A. and E.C. Bain, Principles of Heat Treatment. 5th ed. 1964,

Ohio: American Society for Metals, Metals Park.

70. Judd, R.R. and H.W. Paxton, Kinetics of Austenite Formation from A

Spheroidized Ferrite-Carbide Aggregate. Transaction of the Metallargical

Society, 1968. 242: p. 206-214.

71. Karlsson, B. and L. L.E., Homogenization by Two-Phase diffusion. Materials

Science and Engineering, 1975. 20: p. 161-170.

72. Larsson, L.E. and K. B., Homogenization by One-Phase Diffusion. Materials

Science and Engineering, 1975. 20: p. 155-160.

73. Akbay, T., R. R.C., and A. C., Modelling Reaustenitisation from

Ferrite/cementite Mixtures in Fe-C steels. Acta Metallurgica, 1994. 42: p. 1469-

1480.

74. Atkinson, C. and T. Akbay, The Effect of the Concentration-dependent

Diffusivity of Carbon in Austenite on a Model of Reaustenitisation from

Ferrite/cementite Mixtures in Fe-C Steels. Acta Materialia, 1996. 44: p. 2861-

2868.

75. Jacot, A. and M. Rappaz, A Two-dimensional Diffusion Model for the Prediction

of Phase Transformations: Application to Austenitization and Homogenization of

Hypoeutectoid Fe-C steels. Acta Materialia, 1997. 45: p. 575-585.

76. Jacot, A., M. Rappaz, and R.C. Reed, Modelling of Reaustenitization from the

Pearlite Structure in Steel. Acta Materialia, 1998. 46: p. 3949-3962.

77. Reed, R.C., et al., Determination of Reaustenitization Kinetics in A Fe-0.4C

Steel Using Dilatometry and Neutron Diffraction. Materials Science and

Engineering, 1998. A256: p. 152-165.

Page 158: modelling of phase transformation in hot stamping of boron steel

158

78. Jacot, A. and M. Rappaz, A Combined Model for the Description of

Austenitization, Homogenization and Grain Growth in Hypoeutectoid Fe-C

Steels During Heating. Acta Materialia, 1999. 47: p. 1645-1651.

79. Christian, J.W., The Theory of Transformations in Metals and Alloys. 2ed ed.

1975, Oxford, UK: Pergamon Press Ltd.

80. Roberts, G.A. and R.F. Mehl, The mechanism and the rate of formation of

austenite from ferrite- cementite aggregates. Trans. ASM, 1943. 31: p. 613.

81. Samuels, L.E., Microscopy of Carbon Steels. 1980, American society for Metals,

Metals Park: Ohio, USA.

82. Leblond, J.B. and J. Devaux, A New Kinetic Model for Anisothermal

Metallurgical Transformations in Steels Including Effect of Austenite Grain Size.

Acta metal, 1984. 32: p. 137.

83. Roberts, G.A. and R.F. Mehl, The Mechanism and the Rate of Formation of

Austenite from Ferrite-cementite Aggregates. Transactions of the ASM, 1943.

31: p. 613-650.

84. Saito, Y., Modelling of Microstructural Evolution in Thermomechanical

processing of Structural Steels. Materials Science and Engineering, 1997.

A223: p. 134-145.

85. Bain, E.C., Sorby Centennial Symposium on the History of Metallurgy, ed.

C.S.Smith. 1963, New York: Gordon and Breach Publisher.

86. Smith, C.S., A History of Metallography. 1960: University of Chicago Press.

87. Hehemann, R.F., Phase transformations. 1970, Ohio: American Society for

Metals, Metals Park.

88. Steven, W. and A.J. Haynew, JISI, 1956. 183: p. 349-359.

89. Heheman, R.F., Metals Handbook. 8 ed. Vol. 8. 1973: American Society for

Metals.

90. Takahashi, M., Recent progress: kinetics of the bainite transformation in steels.

Current Opinion in Solid State and Materials Science, 2004. 8(3-4): p. 213-217.

91. Chang, L.C. and H.K.D.H. Bhadeshia, Microstructure of Lower Bainite Formed

at Large Undercoolings Below Bainite Start Temperature. Materials Science

and Technology, 1996. 12: p. 233-236.

92. Bhadeshia, H.K.D.H., Bainite in Steels. 2 ed. 2001, London: Communications

Ltd.

93. Aaronson, H.I., et al., Bainite Viewed Three Different Ways. Metallurgical

Transactions A, 1990. 21(6): p. 1343-1380.

94. Bhadeshia, H.K.D.H., The bainite transformation: unresolved issues. Materials

Science and Engineering A, 1999. 273-275: p. 58-66.

Page 159: modelling of phase transformation in hot stamping of boron steel

159

95. Ohmori, Y. and T. Maki, Bainitic Transformation in View of Displacive

Mechanism. Materials Transactions, JIM, 1991. 32(8): p. 631-641.

96. Ko, T. and S.A. Cottrell, J Iron Steel Inst, 1952. 172: p. 307.

97. Muddle, B.C. and J.F. Nie, Formation of bainite as a diffusional-displacive

phase transformation. Scripta Materialia, 2002. 47(3): p. 187-192.

98. Muddle, B.C., J.F. Nie, and G.R. Hugo, Application of the Theory of Martensite

Crystallography to Displacive Phase Transformations in Substitutional

Nonferrous Alloys. Metallurgical and Materials Transactions A, 1994. 25(9): p.

1841-1856.

99. Christian, J.W., Lattice correspondence, atomic site correspondence and shape

change in "diffusional-displacive" phase transformations. Progress in materials

science, 1997. 42(1-4): p. 101-108.

100. Muddle, B.C. and J.F. Nie, Solid-solid Phase Transform. Japan Institute of

Metals, 1999: p. 1128.

101. Tszeng, T.C. and G. Shi, A Global Optimization Technique to Identify Overall

Transformation Kinetics Using Dilatometry Data - Applications to Austenitization

of Steels. Materials Science and Engineering, 2004. A380: p. 123-136.

102. Maazi, N. and N. Rouag, Modelling Grain Growth in the Presence of Zener

Drag: Application for Fe-3% Si. Modelling Simul. Materials Science and

Engineering, 2001. 9: p. 423-431.

103. Haworth, W.L. and J.G. Parr, The Effect of Rapid Heating on the Alpha-Gamma

Transformation in Iron. Transactions of the American Society for Metals, 1965.

58: p. 476-488.

104. Danon, A., et al., Heterogeneous Austenite Grain Growth in 9Cr Martensitic

Steels: Influence of the Heating Rate and the Austenitization Temperature.

Materials Science and Engineering, 2003. A348: p. 122-132.

105. Siller, R.H. and R.B. Mclellan, The Application of First Order Mixing Statistics to

the Variation of the Diffusivity of Carbon in Austenite. Metallurgical transactions,

1970. 1: p. 985-988.

106. Bhadeshia, H.K.D.H., The Diffusion of Carbon in Austenite. Metal Science,

1981. 15: p. 477-479.

107. Siller, R.H. and R.B. Mclellan, The Variation with Composition of the Diffusivity

of Carbon in Austenite. Transactions of the Metallurgical Society of the

American Institute of Mining, Metallurgical and Petroleum, 1969. 245: p. 697-

700.

108. Capdevila, C., C. García de Andrés, and F.G. Caballero, Incubation Time of

Isothermally Transformed Allotriomorphic Ferrite in Medium Carbon Steels.

Scripta Materialia, 2001. 44.

Page 160: modelling of phase transformation in hot stamping of boron steel

160

109. Karlsson, B. and Z. Metallkd, Transformation Kinetics for the Formation of

Austenite from a Ferritic-Pearlitic Structure. Zeitschrift Fur Metallkunde, 1972.

63(3): p. 160.

110. Smithells, C.J., Metals reference book. Vol. 2. 1967, London: Butterworths.

111. Bhadeshia, H.K.D.H. and J.W. Christian, Bainite in Steels. Metallurgical

Transactions A, 1990. 21A: p. 767-797.

112. Bhadeshia, H.K.D.H., Bainite: overall transformation kinetics. Journal de

physique colloque C4, 1982. 43: p. 443-448.

113. Rees, G.I. and H.K.D.H. Bhadeshia, Bainite Transformation Kinetics Part 1

Modified model. materials Science and Technology, 1992. 8: p. 985-993.

114. Chester, N.A. and H.K.D.H. Bhadeshia, Mathematical Modelling of Bainite

Transformation Kinetics. J. Phys. Ⅳ France, Colloque C5,, 1997. 7: p. 41-46.

115. Quidort, D. and Y.J.M. Brechet, Isothermal growth kinetics of bainite in 0.5% C

steels. Acta Materialia, 2001. 49(20): p. 4161-4170.

116. Hillert, M. and G.R. Purdy, On the misuse of the term bainite. Scripta Materialia,

2000. 43(9): p. 831-833.

117. Lange, W., M. Enomoto, and H. Aaronson, THE KINETICS OF FERRITE

NUCLEATION AT AUSTENITE GRAIN-BOUNDARIES IN FE-C ALLOYS.

Metallurgical transactions. A, Physical metallurgy and materials science, 1988.

19(3): p. 427-440.

118. Bosze, W.P. and R. Trivedi, KINETIC EXPRESSION FOR GROWTH OF

PRECIPITATE PLATES. Metallurgical transactions, 1974. 5(2): p. 511-512.

119. Quidort, D. and Y.J.M. Brechet, A model of isothermal and non isothermal

transformation kinetics of bainite in 0.5% C steels. Isij International, 2002. 42(9):

p. 1010-1017.

120. Crank, J., The Mathematics of Diffusion. 1956, Oxford: Clarendon Press.

121. Avrami, M., Kinetics of Phase Change. Ⅱ. Transformation-time Relations for

Random Districution of Nuclei. J. Chem. Phys., 1940. 8: p. 212-224.

122. Johnson, W.A. and R.F. Mehl, Reaction Kinetics in Processes of Nucleation

and Growth. Trans. Amer. Inst. Min. Met. Eng., 1939. 135: p. 416-443.

123. Kolmogorov, A.N., Statistical Theory of Nucleation Processes. Izv Acad Nauk

SSSR Ser Math, 1937. 3: p. 355-366.

124. Avrami, M., Kinetics of Phase Change Ⅰ. General Theory. J. Chem. Phys.,

1939. 7: p. 1103-1112.

125. Avrami, M., Kinetics of Phase Change. Ⅲ. Granulations, Phase Change, and

Microstructure. J. Chem. Phys., 1941. 9: p. 177-184.

126. Kaufman, L., S.V. Radcliffe, and M. Cohen, Decomposition of Austenite by

Diffusional Processes. 1962, New York: Interscience.

Page 161: modelling of phase transformation in hot stamping of boron steel

161

127. Quidort, D. and Y. Brechet, The role of carbon on the kinetics of bainite

transformation in steels. Scripta Materialia, 2002. 47(3): p. 151-156.

128. Shipway, P.H. and H. Bhadeshia, THE EFFECT OF SMALL STRESSES ON

THE KINETICS OF THE BAINITE TRANSFORMATION. Materials Science and

Engineering a-Structural Materials Properties Microstructure and Processing,

1995. 201(1-2): p. 143-149.

129. Shipway, P. and H.K.D.H. Bhadeshia, Mechanical stabilisation of bainite

Materials Science and Technology, 1995. 11: p. 1116-1128.

130. Singh, S.B. and H.K.D.H. Bhadeshia, Quantitative Evidence for Mechanical

Stabilisation of Bainite. Materials Science and Technology, 1996. 12: p. 610-

612.

131. Chang, L.C. and H. Bhadeshia, Stress-affected transformation to lower bainite.

Journal of Materials Science, 1996. 31(8): p. 2145-2148.

132. Khlestov, V.M., E.V. Konopleva, and H.J. McQueen, Kinetics of austenite

transformation during thermomechanical processes. Canadian Metallurgical

Quarterly, 1998. 37(2): p. 75-89.

133. Russell, K.C., Solids in Phase Transformations. 1968, Ohio: American Society

for Metals.

134. Porter, D.A. and K.E. Easterling, Phase transformations in metals and alloys.

2nd ed. 1992, London: Chapman & Hall.

135. Doremus, R.H., Rates of Phase Transformations. 1985, Orlando: Academic

Press Inc.

136. Li, B., J. Lin, and X. Yao, A novel evolutionary algorithm for determining unified

creep damage constitutive equations. International Journal of Mechanical

Sciences, 2002. 44(5): p. 987-1002.

137. Cao, J. and J. Lin, A study on formulation of objective functions for determining

material models. International Journal of Mechanical Sciences, 2008. 50(2): p.

193-204.

138. Lin, J., T. Dean, and J. Cao, An implicit unitless error and step-size control

method in integrating unified viscoplastic/creep ODE-type constitutive equations.

International journal for numerical methods in engineering, 2008. 73(8): p.

1094-1112.

139. Friedman, P.A. and A.K. Ghosh, Microstructural Evolution and Superplastic

Deformation Behavior of Fine Grain 5083Al. Metall. Mater. Trans. A, 1996.

27A(12): p. 3827-3839.

140. Nicolaou, P.D. and S.L. Semiatin, High-Temperature Deformation and Failure of

an Orthorhombic Titanium Aluminium Sheet Material. Metall. Mater. Trans. A.,

1996. 27A(11): p. 3675-3671.

Page 162: modelling of phase transformation in hot stamping of boron steel

162

141. Davis, J.R., Tensile Testing. 2 ed. 2004, Ohio: ASM International.

142. Spies, M. and K. Salama, Relationship Between Elastic Anisotropy and Texture

in Metal-matrix Composites. Ultrasonics, 1990. 28(6): p. 370-374.

143. Lange, K., Umformtechnik Band 3. 1988, Berlin: Springer.

144. Kocks, U.F., Texture and anisotropy : preferred orientations in polycrystals and

their effect on materials properties. 1998, Cambridge: Cambridge University

Press.

145. Turetta, A., S. Bruschi, and A. Ghiotti. Anisotropic and Mechanical Behavior of

22MnB5 in Hot Stapming Operations. in 10th ESAFORM Conference on

Material Forming. 2007. Zaragoza, Spain: American Institute of Physics.

146. Merklein, M., et al., Mechanical Properties and Plastic Anisotropy of the

Quenchenable High Strength Steel 22MnB5 at Elevated Temperatures. Key

Engeering Materials, 2007. 344: p. 79-86

147. Coghlan, W.A. and W.D. Nix, CONTRIBUTIONS TO THEORY OF JOGGED

SCREW DISLOCATION GLIDE. Metallurgical transactions, 1970. 1(7): p. 1889.

148. Narutani, T. and J. Takamura, GRAIN-SIZE STRENGTHENING IN TERMS OF

DISLOCATION DENSITY MEASURED BY RESISTIVITY. Acta Metallurgica Et

Materialia, 1991. 39(8): p. 2037-2049.

149. Yoshinaga, H., High-Temperature Deforamtion Mechanisms in Metals and

Alloys. Materials Transactions, JIM, 1993. 34(8): p. 635-645.

150. Norton, F.H., The Creep of Steel at High Temperatures. 1929, New York:

McGraw-Hill.

151. Lemaitre, J. and J.-L. Chaboche, Mechanics of solid materials. 1994,

Cambridge: Cambridge University Press.

152. Rabotnov, Y.N., Creep Problems in Structural Members. 1969, Amsterdam:

North-Holland Publishing Company.

153. Lin, J. and T.A. Dean, Modelling of microstructure evolution in hot forming using

unified constitutive equations. Journal of Materials Processing Technology,

2005. 167(2-3): p. 354-362.

154. Cheong, B.H., Modelling of Microstructural and Damage Evolution in

Superplastic Forming, in Munufacturing and Mechanical Engineering. 2002,

University of Birmingham: Birmingham.

155. Foster, A.D., Modelling Damage Evolution During the Hot Deformation of Free

Machining Steels, in Manufacturing and Mechanical Engineering 2007,

University of Birmingham: Birmingham.

156. Dunne, F.P.E., D.R. Hayhurst, and J. Lin, Physically-based temperature

dependence of elastic-viscoplastic constitutive equations for copper between 20

and 500C. Philosophical Magazine A, 1996. 74: p. 359-382.

Page 163: modelling of phase transformation in hot stamping of boron steel

163

157. Lin, J., et al., Development of dislocation-based unified material model for

simulating microstructure evolution in multipass hot rolling. Philosophical

magazine, 2005. 85(18): p. 1967-1987.

158. Nes, E., Modelling of work hardening and stress saturation in FCC metals.

Progress in materials science, 1997. 41(3): p. 129-193.

159. Krausz, A.Z. and K. Krausz, Unified constitutive laws of plastic deofrmation.

Dislocation-density-related Constitutive Modeling, ed. Y. Estrain. 1996, London:

Academic Press.

160. Sandström, R. and R. Lagneborg, A controlling factor for dynamic

recrystallisation. Scripta Metallurgica, 1975. 9(1): p. 59-65.

161. Sandström, R. and R. Lagneborg, A model for hot working occurring by

recrystallization. Acta Metallurgica, 1975. 23(3): p. 387-398.

162. Estrin, Y., Dislocation theory based constitutive modelling: foundations and

applications. Journal of Materials Processing Technology, 1998. 80-1: p. 33-39.

163. Mecking, H. and U.F. Kocks, Kinetics of flow and strain-hardening. Acta

Metallurgica, 1981. 29(11): p. 1865-1875.

164. Lin, J. and Y. Liu, A set of unified constitutive equations for modelling

microstructure evolution in hot deformation. Journal of Materials Processing

Technology, 2003. 143: p. 281-285.

165. Kocks, U., LAWS FOR WORK-HARDENING AND LOW-TEMPERATURE

CREEP. Journal of engineering materials and technology, 1976. 98(1): p. 76-85.

166. Garrett, R.P., J. Lin, and T. Dean, An Investigation of the Effects of Solution

Heat Treatment on Mechanical Properties for AA 6xxx Alloys: Experimentation

and Modelling. International journal of plasticity, 2005. 21: p. 1640-1657.

167. Lin, J., et al., Modelling mechanical property recovery of a linepipe steel in

annealing process. International journal of plasticity, 2009. 25(6): p. 1049-1065.

168. Krajcinovic, D., Damage mechanics: accomplishments, trends and needs.

International Journal of Solids and Structures, 2000. 37(1-2): p. 267-277.

169. Ibijola, E.A., On some fundamental concepts of Continuum Damage Mechanics.

Computer Methods in Applied Mechanics and Engineering, 2002. 191(13-14): p.

1505-1520.

170. Lemaitre, J. and J.-L. Chaboche, Mechanics of solid materials. 1990,

Cambridge: Cambridge University Press.

171. Hayhurst, D.R., Creep Rupture under Multi-axial States of Stress. J. Mech.

Phys. Solids, 1972. 20: p. 381-390.

172. Chaboche, J.L., Continuum Damage Mechanics: Present State and Future

Trends. Nuclear Engineering and Design, 1987. 105: p. 19-33.

Page 164: modelling of phase transformation in hot stamping of boron steel

164

173. Lin, J., F.P.E. Dunne, and D.R. Hayhurst, Aspects of Testpiece Design

Responsible for Errors in Cyclic Plasticity Experiments. Int. J. of Damage

Mechanics, 1999. 8(2): p. 109-137.

174. Bellenger, E. and P. Bussy, Plastic and Viscoplastic Damage Models with

Numerical Treatment for Metal Forming Processes. J. of Materials Processing

Technology, 1998. 80(591-596).

175. Li, Z., B. Bilby, and I. Howard, A study of the internal parameters of ductile

damage theory. Fatigue & fracture of engineering materials & structures, 1994.

17(9): p. 1075-1087.

176. Brust, F.W. and B.N. Leis, A New Model for Characterizing Primary Creep

Damage. Int. J. Fract., 1992. 54: p. 45-63.

177. Tvergaard, V., Material Failure by Void Growth to Coalescence. Adv. App.

Mech., 1990. 27: p. 83-147.

178. Lemaitre, J., How to Use Damage Mechanics. Nuclear Engineering and Design,

1984. 80: p. 233-235.

179. Rice, J.R. and D.M. Tracey, On the ductile enlargement of voids in triaxial

stress fields*. Journal of the Mechanics and Physics of Solids, 1969. 17(3): p.

201-217.

180. Boyer, J.C., E. Vidalsalle, and C. Staub, A Shear Stress Dependent Ductile

Damge Model. J. of Materials Processing Technology, 2002. 121: p. 87-93.

181. Gelin, J.C., Modelling of Damage in Metal Forming Processes. J. of Materials

Proc. Tech, 1998. 80: p. 24-32.

182. Semiatin, S.L., et al., Cavitation during Hot Tension Testing of Ti-1Al-4V.

Materials Science and Engineering 1998. A256: p. 92-110.

183. Dyson, B.F. and M. McLean, Particle-coarsening , σ0 and Tertiary Creep. Acta.

Met., 1983. 30: p. 17-27.

184. Lin, J., T. Dean, and Y. Liu, A review on damage mechanisms, models and

calibration methods under various deformation conditions. International journal

of damage mechanics, 2005. 14(4): p. 299-319.

185. Dyson, B.F., Physically-based Models of Metal Creep for Use in Engineering

Design, in Modelling of Materials Behaviour and Design, J.D. Embury and A.W.

Thompson, Editors. 1990, The Materials, Metals and Materials Society: London.

p. 59-75.

186. Lin, J., D.R. Hayburst, and B.F. Dyson, The Standard Ridges Uniaxial

Testpiece: Computed Accuracy of Creep Strain. J. of Strain Analysis, 1993.

28(2): p. 101-115.

Page 165: modelling of phase transformation in hot stamping of boron steel

165

187. Ashby, M.F. and B.F. Dyson, Creep Damage Mechanics and Micro-

mechanisms, in ICF Advance in Fracture Research. 1984, Pergamon Press. p.

3-30.

188. Cocks, A.C.F. and M.F. Ashby, On creep fracture by void growth. Progress in

materials science, 1982. 27(3-4): p. 189-244.

189. Vetrano, J.S., E.P. Simonen, and S.M. Bruemmer, Evidence for Excess

Vacancies at Sliding Grain Boundaries During Superplastic Deformation. Acta

Metallia, 1999. 47(15): p. 4125-4149.

190. Raj, R. and M.F. Ashby, Intergranular Fracture at Elevated Temperatrues. Acta

Metallurgica, 1975. 23(653-666).

191. Ridley, N., Cavitations and Superplasticity, in Superplasticity, AGARD Lecture

Series No. 168. 1989, Essex: Specialised Printing Services Limited. p. 4.1-4.14.

192. Chokshi, A.H. and T.G. Langdon, A Model for Diffusional Cavity Growth in

Superplasticity. Acta Metallurgica, 1987. 35(1089-1101).

193. Lin, J., B.H. Cheong, and X. Yao, Universal Multi-objective Function for

Opimizing Superplastic Damage Constitutive Equations. J. of Materials Proc.

Tech, 2002. 125: p. 199-205.

194. Khaleel, M.A., H.M. Zbib, and E.A. Nyberg, Constitutive Modelling of

Deformation and Damage in Superplastic Materials. Int. J. of Plasticity, 2001.

17(177-296).

195. Zheng, M., et al., A ductile damage model corresponding to the dissipation of

ductility of metal. Engineering Fracture Mechanics, 1996. 53(4): p. 653-659.

196. Odqvist, F.K.G. and J. Hult, Some Aspects of Creep Rupture. Atk. Fys., 1961.

19(26): p. 379-382.

197. Dyson, B., CREEP AND FRACTURE OF METALS - MECHANISMS AND

MECHANICS. Revue de physique appliquée, 1988. 23(4): p. 605-613.

198. Khlestov, M.V., V.E. Konopleva, and H.J. Mcqueen, Kinetics of austenite

transformation during thermomechanical processes. Cnadian Metallurgical

Quarterly, 2008. 37(2): p. 75-89.

199. Drozdov, B.Y., L.I. Kogan, and R.I. Entin, The physics of metals and

metallography, 1962. 13(5): p. 135-138.

200. Mutiu, T.A., A.J. Kinderman, and I.M. Bernstein, The Hot Deformation of

Austenite, ed. J.B. Ballance. 1977, New York: T.M.S.-A.I.M.E. 410-427.

201. Cumming, D.M., et al., Multiple Material Bumper Beam. 2005, Magna

International Inc.: United States.

202. Thomas, D. and D.T. Detwiler, Microstructural Optimization of Automotive

Structures. 2009, HONDA MOTOR CO., LTD.: United States.

Page 166: modelling of phase transformation in hot stamping of boron steel

166

203. Guiles, M., et al., Selectively Annealed Bumper Beam. 2008, Shape

Corporation: United States.

204. Bodin, H., Method of Hot Stamping and Hardening A Metal Sheet. 2010,

Gestamp Hardtech AB: United States.

Page 167: modelling of phase transformation in hot stamping of boron steel

167

Appendix

Appendix 1 Development of austenite at different temperatures (700℃, 770℃, 840℃,

800℃ and 925℃) when heating rates are 1.25℃/s and 5.0℃/s.

a) Temperature=700℃ & Heat rate=5.0℃/s.

b) Temperature=770℃ & Heat rate=5.0℃/s.

c) Temperature=840℃ & Heat rate=5.0℃/s.

Page 168: modelling of phase transformation in hot stamping of boron steel

168

d) Temperature=900℃ & Heat rate=5.0℃/s.

e) Temperature=800℃ & Heat rate=1.25℃/s.

Page 169: modelling of phase transformation in hot stamping of boron steel

169

Appendix 2 Strain to failure of tensile test samples of different gauge length. The

forming temperature is 700℃. Strain rates are 0.1/s, 1.0/s and 10.0/s.

0.1/s

//s

10.0/s

//s

1.0/s

//s

Page 170: modelling of phase transformation in hot stamping of boron steel

170

Appendix 3 Stress strain relationship of test pieces experienced different soaking

conditions (870℃ for 1 minute, 925℃ for 5 minutes and 950℃ for 10 minutes).

5min-925℃-700℃forming

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4

strain

str

es

s(M

Pa

)

1min-870℃-700℃forming

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4

strain

str

es

s(M

Pa

)

10min-950℃-700℃forming

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4

strain

str

es

s(M

Pa

)

Page 171: modelling of phase transformation in hot stamping of boron steel

171

Appendix 4 Mechanical responses of USIBOR 1500P during Gleeble tension tests (to

failure). a) Temperature=700℃, strain rates are in a range of 0.01s-1 to 10s-1; b) Strain

rate =0.1/s, the temperatures are from 500℃ to 800℃.

a)

b)

Temperature=700℃,

0

50

100

150

200

250

300

350

0 0.1 0.2 0.3 0.4 0.5 0.6

Strain

Str

ess (

MP

a) 0.01/s

0.1/s

1.0/s

10.0/c

10.0/s

1.0/s

0.1/s

0.01/s

Strain rate=0.1/s

0

100

200

300

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Strain

Str

ess (

MP

a)

700

℃800

℃500

℃600

700℃

800℃

600℃

Strain rate=0.1/s