Farideh_HajyAkbary_Final_Thesis_Print.

Farideh_HajyAkbary_Final_Thesis_Print.
Optimising mechanical behaviour of new advanced steels
based on fine non-equilibrium microstructures
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op
[09, 12, 2015] om [10:00] uur
door
Farideh HAJYAKBARY
Master of Science in Metallurgy and Materials Engineering, University of Tehran,
Tehran, Iran
Geboren te Qom, Iran.
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. J. Sietsma
Copromotor:
Dr. M. J. Santofimia
Samenstelling promotiecommissie:
Rector Magnificus,
voorzitter
Prof. dr. ir. J. Sietsma,
Technische Universiteit Delft, the Netherlands, promotor
Dr. M. J. Santofimia,
Technische Universiteit Delft, the Netherlands, copromotor
Dr. G. Miyamoto,
Tohuku University, Tohoku, Japan
Onafhankelijke leden:
Prof. dr. I.M. Richardson,
Technische Universiteit Delft, the Netherlands
Prof. dr. ir. L.A.I. Kestens,
Ghent University, Ghent, Belgium
Prof. dr. Kip Findley,
Colorado School of Mines, Colorado, USA
Dr. D. S. van Bohemen,
Tata Steel Research Development and Technology, IJmuiden,
the Netherlands
This research was carried out under the project number M41.10.11437 in the framework of
the Research Program of the Materials innovation institute (M2i) in the Netherlands
(www.m2i.nl).
ISBN 978-94-6295-398-7
Copyright © 2015, Farideh HajyAkbary
[email protected]
All rights reserved. No part of the material protected by this copy right notice may be
reproduced or utilized in any form or by any means, electronically, including photocopy,
recording or by any information storage and retrieval system, without permission from the
author.
Printed by: Proefschriftmaken.nl
This thesis is dedicated to
my little angel Arshida
Contents
TABLE OF CONTENTS
1
Introduction ------------------------------------------------------------------------------------------------ 1
1.1
1.2
1.3
2
Research aims -------------------------------------------------------------------------------------------------------------------- 3
Content of the thesis------------------------------------------------------------------------------------------------------------ 3
References ------------------------------------------------------------------------------------------------------------------------- 5
Effects of specimen size on the tensile behavior of steels -------------------------------------- 7
2.1
2.2
2.3
2.4
2.5
2.6
Introduction ----------------------------------------------------------------------------------------------------------------------- 8
Mathematical modelling of the effective parameters on the crosshead displacement --------------------- 10
Experimental procedure ------------------------------------------------------------------------------------------------------ 12
Results and discussion --------------------------------------------------------------------------------------------------------- 15
Conclusions ----------------------------------------------------------------------------------------------------------------------- 23
References ------------------------------------------------------------------------------------------------------------------------ 24
3 An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density in
Lath Martensitic Structures --------------------------------------------------------------------------------- 27
3.1
3.2
3.3
3.4
3.5
3.6
Introduction ---------------------------------------------------------------------------------------------------------------------- 28
Calculation of dislocation density ------------------------------------------------------------------------------------------ 29
Experimental procedure ------------------------------------------------------------------------------------------------------ 34
Results and discussion --------------------------------------------------------------------------------------------------------- 36
Conclusions ----------------------------------------------------------------------------------------------------------------------- 47
References ------------------------------------------------------------------------------------------------------------------------ 48
4 Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Quenching and
Partitioning Steel ---------------------------------------------------------------------------------------------- 51
4.1
4.2
4.3
4.4
4.5
4.6
Introduction ---------------------------------------------------------------------------------------------------------------------- 52
Experimental procedures ----------------------------------------------------------------------------------------------------- 53
Results ----------------------------------------------------------------------------------------------------------------------------- 53
Discussion ------------------------------------------------------------------------------------------------------------------------- 63
Conclusions ----------------------------------------------------------------------------------------------------------------------- 68
References ------------------------------------------------------------------------------------------------------------------------ 69
5 Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Quenching and
Partitioning Steel ---------------------------------------------------------------------------------------------- 71
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Introduction ---------------------------------------------------------------------------------------------------------------------- 72
Theoretical calculation of the yield strength of the constituent phases ----------------------------------------- 72
Experimental procedure ------------------------------------------------------------------------------------------------------ 74
Results ----------------------------------------------------------------------------------------------------------------------------- 77
Discussion ------------------------------------------------------------------------------------------------------------------------- 88
Conclusions ----------------------------------------------------------------------------------------------------------------------- 92
References ------------------------------------------------------------------------------------------------------------------------ 93
6 Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenching and Partitioning
Steel --------------------------------------------------------------------------------------------------------------- 95
6.1
6.2
6.3
6.4
6.5
7
Introduction ---------------------------------------------------------------------------------------------------------------------- 96
Experimental procedure ------------------------------------------------------------------------------------------------------ 96
Results and discussion --------------------------------------------------------------------------------------------------------- 98
Conclusion ---------------------------------------------------------------------------------------------------------------------- 101
References ---------------------------------------------------------------------------------------------------------------------- 102
Conclusions and recommendations -------------------------------------------------------------- 103
7.1
7.2
Conclusions --------------------------------------------------------------------------------------------------------------------- 104
Recommendations for future research --------------------------------------------------------------------------------- 105
iv
Contents
Summary---------------------------------------------------------------------------------------------------------107
Samenvatting---------------------------------------------------------------------------------------------------109
Acknowledgements-------------------------------------------------------------------------------------------113
List of publications---------------------------------------------------------------------------------------------115
About the author----------------------------------------------------------------------------------------------117
CHAPTER 1
1 Introduction
Car industry, as a main consumer of steels, is interested in steels with high strength and
high formability to reduce the fuel consumption and increase passenger safety [1].
Therefore, a significant research effort has been directed towards the development of
Advanced High Strength Steels (AHSS) which have a good combination of high strength and
ductility. The microstructure of AHSS consists of at least two different phases, one hard
phase like martensite or bainite and one soft phase such as ferrite or retained austenite [2].
The AHSS grades that are currently being applied or are under increased investigation by
steel researchers can be categorized into three generations [3], as presented in Fig. ‎1-1. The
first generation of AHSS contains fairly low alloy steels, with a multiphase microstructure
that is primarily ferritic-based. These steels are well established and they are currently the
most applied AHSS which results, apart from their improved strength and formability, from
their low price. This generation includes dual phase (DP) steels, transformation induced
plasticity (TRIP) steels, complex phase (CP) steels and martensitic steels. The second
generation of AHSS have excellent mechanical properties, but they are highly alloyed steels,
resulting in a significant cost increase. This generation involves high strength steels such as
austenitic twinning induced plasticity (TWIP) steels and lightweight steels with induced
plasticity (L-IP) [4]. Academic and industrial researchers are interested to develop the third
generation of AHSS with strength and ductility at the same levels that are exhibited by the
second generation but with lower alloying levels [5]. To develop the third generation of
AHSS, attention has been paid to processes which deliver a hard bainitic or martensitic
matrix containing a dispersion of retained austenite. These microstructural components are
phases that are formed in non-equilibrium conditions and/or remain in the steel
microstructure under metastable conditions [6].
1
Introduction
Fig. ‎1-1 Elongation-strength relationships for different grades of steels [7].
One of the most promising heat treatments for the development of the third generation
of AHSS is the Quenching and Partitioning (Q&P) process. A schematic of the Q&P treatment
is illustrated in Fig. ‎1-2. The Q&P process involves rapid quenching of an austenitic
microstructure to a temperature lower than the martensite-start temperature (Ms) to form a
controlled fraction of martensite. Here, the martensite which is formed during the initial
quenching is called initial martensite. The process is followed by an isothermal treatment,
either at or above the initial quenching temperature. This isothermal holding process is
called partitioning process and it is aimed to allow carbon to partition from supersaturated
martensite to austenite and stabilizes austenite. The treatment is ended by quenching the
microstructure to room temperature [8]. Secondary martensite may form during the final
quenching, if some parts of austenite do not become stable enough [9]. Moreover, bainite
may form by decomposition of austenite during the isothermal treatment [10]. Accordingly,
the Q&P microstructures can be composed of initial martensite, bainite, secondary
martensite and retained austenite and, depending on the final microstructure, varying
ranges of mechanical properties can be achieved. In this matter, the key to optimise the
mechanical properties of these steels and make them commercialized is understanding the
relation of the microstructural and mechanical properties. This can be achieved by
investigation of the contributions of dislocations, precipitations, morphology and chemical
composition of the phases on their independent mechanical behavior as well as studying the
synergistic influence of the phases on the ductility and strength of the Q&P steels.
The Q&P process is a complicated heat treatment and the final microstructure is sensitive
to the exact temperature profile and therefore also to the temperature gradients. At the
laboratory scale, an accurate control of the heating process and avoiding temperature
inhomogeneity within the specimen is possible by heat treating small specimens. The
evaluation of the mechanical properties of these small specimens can be performed by using
microtensile tests. Using the microtensile test for mechanical property characterization
creates concerns about whether the measured mechanical properties are influenced by the
specimen dimensions. Therefore, it is important to study the relation of the results of the
microtensile tests with the performance of the material at a macro-scale.
2
Chapter 1
Fig. ‎1-2 Scheme of the Q&P process. Here, ɣ is initial austenite, M1 is initial martensite, M2 is secondary
martensite, B is bainite, RA is retained austenite, C i is carbon content of the steel, Cɣ is carbon content of
austenite and CM1 is carbon content of initial martensite.
1.1
Research aims
The aims of the research described in this thesis are three-fold:
(1) To correlate small-scale and conventional tensile test methods.
(2) To investigate microstructural development during the Q&P process.
(3) To identify the contributions of (non-equilibrium) microstructural components on the
mechanical properties of the Q&P steels, in order to deliver AHSS with superior mechanical
properties.
1.2
Content of the thesis
In line with the research aims, the thesis is divided into three parts: chapter 2 investigates
the influence of specimen size on the tensile behaviour of steels, chapters 3 and 4 outline
the characterization of the microstructural properties of Q&P microstructures developed in a
0.3C-1.6Si-3.5Mn (wt.%) steel with non-homogenous chemical composition and chapters 5
and 6 discuss the relation between the tensile properties and the microstructural properties.
Chapter 2 studies the influence of the specimen geometry on the tensile behaviour of
steels. This is done by tensile testing of miniature and standard specimens from different
grades of steels. Moreover, a model is developed to determine the elastic strain of the
miniature specimens from the crosshead displacement of the tensile test machine.
Chapter 3 introduces an improved method to measure dislocation density of a lath
martensitic steel by applying an X-ray diffraction profile analysis method. The proposed
method is would choice due to the considered range of the Fourier length. This method
leads to a dislocation density that is in good agreement with the dislocation density
determined based on the dislocation strengthening.
Chapter 4 investigates the relations between the Q&P process parameters, the local
chemical composition and the microstructural properties. A comprehensive microstructural
analysis of the constituent phases, including their volume fractions and chemical
compositions, is performed. The interactions between bainite formation, carbide
3
Introduction
precipitation and carbon partitioning process on the microstructural development is
discussed.
Chapter 5 studies contributions of dislocations, precipitates, morphology and chemical
composition of the constituent phases on their independent yield strength. The influence of
instability of austenite on the yield strength of the Q&P microstructures is investigated by
applying in-situ X-ray diffraction. Moreover, the synergistic influence of the phases on the
ductility and strength of the Q&P microstructures is analysed.
Chapter 6 discusses the key microstructural parameters which result in developing Q&P
microstructures with a good combination of tensile strength and ductility. Furthermore, the
mechanical properties of the developed microstructures are compared with other types of
AHSS.
Chapter 7 summarizes the main conclusions of the project and provides
recommendations for future research.
4
Chapter 1
1.3
References
[1] R. Kuziak, R. Kawalla and S. Waengler, “Advanced high strength steels for automotive
industry”, Archives of Civil and Mechanical Engineering, vol. 8, pp. 103-117, 2008.
[2] E. De Moor, P. J. Gibbs, J. G. Speer, D. K. Matlock and J. G. Schroth, “Strategies for third
generation advanced high-strength steel development”, AIST Transactions, Iron & Steel
Technology Transactions, vol. 7, pp. 133-144, 2010.
[3] S. Keeler and P. Ulintz, “Advanced high strength steels solve glowing demands for
formability”, Met. Form., vol. 45, pp. 24-28, 2011.
[4] L. Samek, E. Arenholz, R. Schneider and J. Gentil, “Influence of the thermal processing on
the microstructure and mechanical properties of a high-performance high-manganese
steel”, Metal Conference, Czech Republic, 2012.
[5] A. Grajcar, R. Kuziak and W. Zalecki, “Third generation of AHSS with increased fraction of
retained austenite for the automotive industry”, Archives of Civil and Mechanical
Engineering, vol. 12, pp. 334-341, 2012.
[6] D. K. Matlock and J. G. Speer, “Design considerations for the next generation of advanced
high strength sheet steels”, The Conference of Korean Institute of Metals and Materials,
Korea, 2006.
[7] J. N. Hal, “Evolution of Advanced High Strength Steels in Automotive Applications”, Great
Design in Steels Seminar, 2011.
[8] D. V. Edmonds, K. He, F. C. Rizzo, B. C. De Cooman, D. K. Matlock and J. G. Speer,
“Quenching and partitioning martensite—A novel steel heat treatment”, Mater. Sci. Eng. A,
vol. 438–440, pp. 25–34, 2006.
[9] J. Mola and B. C. De Cooman, “Quenching and Partitioning (Q&P) processing of
martensitic stainless steels”, Matal. Mater. Trans. A, vol. 44, pp. 946-967, 2013.
[10] A. J. Clarke, J. G. Speer, M. K. Miller, R. E. Hackenberg, D. V. Edmonds, D. K. Matlock, F.
C. Rizzo, K. D. Clarke and E. De Moor, “Carbon partitioning to austenite from martensite or
bainite during the quench and partition (Q&P) process: A critical assessment”, Acta Materal.,
vol. 56, pp. 16-22, 2008.
5
Introduction
6
CHAPTER 2
2 Effects of specimen size on the tensile
behavior of steels*
Abstract
The effect of the specimen’s parallel length on the tensile behavior of four different
grades of steels was studied. The steels that were object of this analysis were one interstitial
free steel, two dual phase steels with different fraction of ferrite and one martensitic steel.
Miniature specimens were tested in two different geometries, with parallel lengths of 4 mm
and 3 mm. The measurement of elastic strain in the miniature specimens was done by
means of the crosshead displacement of the tensile-test machine. Since the elastic
elongation of the fillet zones of the tensile specimen and machine compliance were
recorded along with the elastic elongation of the tensile specimen as the crosshead
displacement, measuring the elastic strain by this method led to an overestimation of strain.
A mathematical model for calculating the elastic elongation of the fillet-zones of a dog-bone
tensile specimen and the machine compliance as a function of the applied load was
proposed. The subtraction of the fillet-zones elongation and the machine compliance from
the crosshead displacement allowed the calculation of the elastic elongation of miniature
specimens, leading to values in agreement with strains measured via digital image
correlation. Comparing the tensile behaviour of the miniature specimens and A80 standard
specimens showed that reducing the specimen parallel length did not influence the observed
yield stress and tensile strength of the steel. The fracture strain of the miniature specimens
was higher than of the standard ones. A correction method was applied to correct the
fracture strain of the miniature specimens.
Keywords: Micro-tensile test, Crosshead displacement, Fillet-zones, Machine compliance
Yield strength, Tensile strength, Fracture strain.
*
This chapter is based on a scientific paper:
F. HajyAkbary, M. J. Santofimia and J. Sietsma, Elastic strain measurement of miniature tensile specimens,
Experimental Mechanics, vol. 54, pp. 165-173, 2014.
7
Effects of specimen size on the tensile behavior of steels
2.1
Introduction
Nowadays, the development of new fabrication technologies and miniaturized products
which restrict the specimen dimensions results in an increased use of miniaturized tests for
studying mechanical properties of materials. Additionally, analysing miniature specimens
instead of standard ones saves material and time for both industrial and academic
researchers.
Generally, tensile specimens used in miniaturized tests are dog-bone shaped. A dog-bone
tensile specimen can be divided into five zones: the parallel-zone, the two fillet-zones and
the two grip-zones as shown in Fig. ‎2-1. Dimensions of miniature specimens deviate from
ASTM standards, the parallel length of miniature specimens being in the range from 1 mm
[1] to several millimetres [2, 3]. This situation naturally invites concern as to whether the
geometries/dimensions of the miniature specimens have any influence on the experimental
results and, if so, how strong these influences are [4].
According to an investigation by Zhao et al. [5] reducing the parallel length has no
influence on the yield stress of ultra-fine grained copper, but produces an increase on
ductility. For the case of steel, there is a lack of research considering the influence of the
specimen parallel length on the mechanical behavior during microtensile testing. Therefore
it is essential to do a comprehensive study on the tensile behavior of steels with miniature
dimension.
Another difficulty of application of miniature specimens to characterize the mechanical
properties is that measuring the precise elastic strain is challenging. The elastic strain of
standard specimens can be determined by using clip-on extensometers, but this option is
difficult in miniature specimens due to their small dimensions [6]. At present, there exist
non-contacting strain measuring systems such as laser and video extensometers [7] but their
high price and complex set-up limit their application. Therefore, using extensometers is not a
common method for elastic strain measurements of miniature specimens.
There are two main alternatives to the use of extensometers for determining the elastic
strain of miniature specimens: the application of Digital Image Correlation (DIC) and the
measurement of the strain from the crosshead displacement that is recorded by the tensiletest machine. The DIC method consists of the measurement of the strain of the specimen
during testing by comparing, pixel by pixel, images of the specimen before and after
elongation [8]. This technique requires the use of a high resolution camera, followed by data
post-processing by using the corresponding software. On the other hand, the measurement
of the elastic strain of miniature specimens from the crosshead displacement does not need
any extra equipment and data processing. However, one drawback of using crosshead
displacement for elastic strain measurements is that the elongation of the fillet-zones of the
specimen and the tensile machine parts are included in the crosshead displacement.
Therefore, this method overestimates the elastic strain of miniature specimens, as will be
detailed further on.
8
Chapter 2
Fig. ‎2-1 Scheme of a tensile-test machine and a dog-bone specimen which are modelled by five series of
springs.
Although elongation of the fillet-zones of miniature specimens during tensile tests has
been reported by different researchers [5], most of the published investigations considered
the fillet-zones as rigid items when measuring the strain from the crosshead displacement
[9]. To eliminate the influence of the fillet-zones elongation on the measured elastic strain,
Koubaa et al. [10] defined the initial length in the strain calculation as the total length of the
parallel-zone and the two fillet-zones. The strain measured by their proposed approach is in
better agreement with the strain calculated using finite element analysis than the strain
measured by dividing the crosshead displacement by the initial length of the parallel-zone.
However, their proposed method underestimates the strain, since the strain in the filletzones is smaller than in the parallel-zone.
A tensile-test machine is not a monolithic part and it consists of different parts like the
machine frame together with measuring and fixturing devices. The machine components are
not rigid and they deform elastically in tension. These elongations, which are known as
machine compliance, are included in the recorded crosshead displacement [11]. The elastic
elongation of miniature specimens is relatively small and the machine compliance has
significant effect on the crosshead displacement. Therefore, the machine compliance should
be precisely considered when the elastic strain of miniature specimens is to be measured
from the crosshead displacement. The ASTM standard for tensile testing of single filament
materials determines the machine compliance by assuming the tensile-test machine and
specimen as two linear springs which are connected in series [12]. According to this
standard, the machine stiffness depends on the specimen stiffness and dimensions.
However, experimental measurements of the machine stiffness reveal that the machine
stiffness is a function of the applied load and it is independent of the specimen properties
[13].
In this chapter, miniature and standard specimens of four different types of steels were
tested in tension. The influence of the specimen dimension on the yield strength, the tensile
strength and the fracture strain was studied. Furthermore, a new correction method was
established to determine the elastic strain of miniature specimens from the crosshead
displacement. This was done by the calculation of the fillet-zones elastic elongation and the
machine compliance and subtracting their values from the recorded crosshead
displacement. This correction method was used to calculate the elastic strain in the parallelzone of the miniature specimens. Resulting values were in good agreement with the elastic
strains measured from DIC method.
9
Effects of specimen size on the tensile behavior of steels
2.2
Mathematical modelling of the effective parameters on the crosshead displacement
A tensile-test system consists of tensile specimen and tensile-test machine. The elastic
elongation of the tensile-test system components during the tensile test can be considered
as the elongation of a series of springs. In this chapter, the tensile test system is modelled by
five springs in series: two for the tensile-test machine, two for the fillet-zones and one for
the parallel-zone of the tensile specimen (Fig. ‎2-1). In the current approach, each arm of the
tensile-test machine is modelled by a spring with stiffness of 2𝐾𝑚 . A factor 2 is included to
simplify the calculation procedure so the total stiffness of the tensile-test machine can be
considered as a single spring with stiffness of 𝐾𝑚 . The apparent stiffness that is displayed by
the tensile-test system (𝐾𝑎𝑝𝑝 ) is calculated as:
1
1
1
2
=𝐾 +𝐾 +𝐾 ,
𝐾𝑎𝑝𝑝
𝑚
𝑝
‎2-1
𝑓
where 𝐾𝑚 , 𝐾𝑝 and 𝐾𝑓 are the stiffness of the tensile-test machine, the stiffness of the
parallel-zone and the stiffness of one fillet of the fillet-zones of the tensile specimen,
respectively. The total elongation of the tensile-test system is recorded as crosshead
displacement by the tensile-test machine. This recorded displacement is defined here as the
apparent elongation of the tensile specimen (∆𝑙𝑎𝑝𝑝 ) and it can be calculated by:
∆𝑙𝑎𝑝𝑝 = ∆𝑙𝑚 + ∆𝑙𝑝 + 2∆𝑙𝑓 ,
‎2-2
where ∆𝑙𝑚 , ∆𝑙𝑃 and ∆𝑙𝑓 are the elongation of the tensile-test machine parts, the elongation
of the parallel-zone and the elongation of one of the fillet-zones of the tensile specimen,
respectively.
2.2.1 Elastic elongation of the fillet-zones
In this section, a model is developed to calculate the elastic elongation of the fillet-zones.
It is known that the elastic strain can be calculated from the Hooke’s Law:
𝜀=
𝜎
,
𝐸
‎2-3
where ε, σ and E are the elastic strain, the stress and the Young’s modulus of the material,
respectively. The elastic strain of one fillet of the fillet-zones (𝜀𝑓 ) at location x, which is the
distance between the boundary of the grip-zone and the fillet-zone (Fig. ‎2-2), is determined
as:
𝜀𝑓 =
𝐹
,
𝐸𝑑𝑤𝑓 (𝑥)
‎2-4
where F is the applied force and d is the specimen thickness. Here, 𝑤𝑓 (𝑥) is the specimen
width in the fillet-zone as a function of x.
10
Chapter 2
Fig. ‎2-2. Scheme of a fillet-zone of a dog-bone tensile specimen.
The function 𝑤𝑓 (𝑥) is defined as:
𝑤𝑓 (𝑥) = 𝑤𝑔 − 2 √𝑟 2 − (𝑟 − 𝑥)2 ,
‎2-5
where 𝑤𝑔 is the specimen width at the boundary of the fillet-zone and the grip-zone and r is
the fillet-zone radius. The elongation of a fillet-zone is determined as:
𝑟
∆𝑙𝑓 = ∫0 𝜀𝑓 𝑑𝑥 ,
‎2-6
Thereupon, the elongation of a fillet-zone is calculated by substituting Eq. ‎2-4 and Eq. ‎2-5
into Eq. ‎2-6, as:
𝑟
∆𝑙𝑓 = ∫
0
𝐹
𝑑𝑥
×
.
𝐸𝑑 𝑤𝑔 − 2√𝑟 2 − (𝑟 − 𝑥)2
‎2-7
The ratio of the elongation of a fillet-zone to the parallel-zone is:
𝑟
∆𝑙𝑓
∆𝑙𝑝
∫0
=
𝑑𝑥
𝑤𝑔 −2√𝑟2 −(𝑟−𝑥)2
𝑙𝑃
𝑤𝑝
‎2-8
=𝛼,
where 𝑤𝑃 and 𝑙𝑃 are the width and the length of the specimen in the parallel-zone,
respectively. The parameter 𝛼 is a geometrical coefficient and it is independent of the
applied force and the material.
2.2.2
Machine compliance
Elastic elongation of components of a tensile-test machine during the tensile test can be
modelled by the elongation of a spring with stiffness 𝐾𝑚 . It is well established that the
stiffness of a tensile-test machine (𝐾𝑚 ) is a non-linear function of the applied force and it is
independent of the specimen type and geometry [13]. The machine compliance (∆𝑙𝑚 ) is
defined by the Hooke’s Law as:
11
Effects of specimen size on the tensile behavior of steels
∆𝑙𝑚 =
𝐹
,
𝐾𝑚
‎2-9
where F is the applied force. Since 𝐾𝑚 is a function of the applied load, it can be concluded
from Eq. ‎2-9 that the compliance of a tensile-test machine is a function of the applied load.
The compliance function is invariant for different specimens and it can be used to calculate
the machine compliance at a certain value of the applied force.
2.2.3 Method to calculate the parallel-zone strain
The elastic elongation of the parallel-zone can be calculated from the crosshead
displacement by taking the following steps:
a) The first step is specifying the machine compliance function. In this matter, a specimen
is tested by the tensile-test machine while the reference elongation through its parallel𝑟𝑒𝑓
zone, ∆𝑙𝑝 , is recorded by a direct method such as DIC. The machine compliance, at a
certain value of force, is calculated by combining Eq. ‎2-2 and Eq. ‎2-8 and substituting the
corresponding values of the crosshead displacement (∆𝑙𝑎𝑝𝑝 ) and the reference elongation
𝑟𝑒𝑓
(∆𝑙𝑝 ) in:
𝑟𝑒𝑓
∆𝑙𝑚 = ∆𝑙𝑎𝑝𝑝 − (1 + 2𝛼)∆𝑙𝑝 .
‎2-10
The geometrical coefficient 𝛼, is computed by considering the specimen dimensions in
Eq. ‎2-8. Finally, the compliance function (∆𝑙𝑚 ) of the tensile-test machine is determined by
plotting the machine compliance-force (∆𝑙𝑚 𝑣𝑠. 𝐹) diagram.
b) Then, the corrected elongation of the parallel-zone (∆𝑙𝑝𝑐 ) for every tensile specimen at
a certain value of the applied load and apparent elongation is determined by combining
Eq. ‎2-2 and Eq. ‎2-8 and substituting the machine compliance function in the following
equation:
∆𝑙𝑝𝑐 =
1
(∆𝑙𝑎𝑝𝑝 − ∆𝑙𝑚 ).
(1 + 2𝛼)
‎2-11
The corrected elastic strain within the parallel-zone of miniature specimens (𝜀𝑝𝑐 ) is
expressed as:
1
∆𝑙𝑎𝑝𝑝 −∆𝑙𝑚
𝜀𝑝𝑐 = (1+2𝛼) (
𝑙𝑝
).
‎2-12
Equation ‎2-10 is independent of the material type and it can be determined for every
tensile specimen at a certain value of the applied load and apparent elongation.
2.3
Experimental procedure
In this work, the tensile behavior of four different steels: e. g. one interstitial free steel
(IF), two dual phase steels with different fraction of ferrite (DP1000 and DP600) and one
martensitic steel (M1400), was studied using specimens with different sizes and geometries.
The key points of the experimental procedure are given in this section.
12
Chapter 2
Table ‎2-1. Nominal dimensions of the standard and miniature specimens (mm)
Specimen
T.120
T.4
T.3
Width
20.0
1.0
0.8
Thickness
1.0
1.0
0.8
Parallel length
120.0
4.0
3.0
Grip area
30.0×40.0
4.0×2.0
3.5×3.0
Overall length
230.0
10.0
10.0
Fillet radius
15.0
1.0
0.5
Dog-bone miniature specimens with two different dimensions and standard A80 tensile
specimens were tested in tension. The effect of the specimen dimensions on the elastic
elongation was studied by analysing the elastic strain measurement of the miniature
specimens, via DIC, and the standard specimens. Moreover, the elastic strain of the
miniature specimens was measured from the crosshead displacement and the results were
compared with the elastic strain which was measured by DIC. A mathematical model was
developed to correct the elastic strains of the miniature specimens which were measured
from the crosshead displacement. This was done based on the elastic strain measurement of
the miniature specimens from steel M1400. This model included the subtraction of the filletzones elongation and the machine compliance from the crosshead displacement. The
proposed correction method was experimentally validated with miniature specimens from
different types of steels (DP1000, DP600 and IF).
2.3.1 Specimens geometry
The specimen dimensions were listed in Table ‎2-1. In the appellation of the specimens, T
refers to the tensile specimen and the next number shows the specimen parallel length in
millimetre. The dimensions of the miniature specimens satisfied some of the ASTM standard
requirements. The standard indicates that the ratio of the parallel length to the parallel
width was 4 and the radius of the fillet-zone is equal or greater than the width of the
parallel-zone [14]. Also, to ensure that the specimen failure will occur within the parallelzone, the standard specifies a ratio of the grip width to the parallel width equal or higher
than 1.5 [15]. Miniature specimens were machined from sheets using an electro discharge
machine while the specimen axis was perpendicular to the rolling direction.
2.3.2 Tensile testing
Three specimens were tested for each group of geometries and steel, except for the
M1400 steel in which only one specimen was tested due to slippage difficulties. All the
standard and miniaturized tensile tests were done until failure. Standard specimens were
tested with a "Schenk Trebel tensile-test machine 100KN". A uniform elongation region with
initial length of 80 mm was considered as the gauge length of the standard specimens. The
strain of the standard specimens (εsp ), within the gauge length, was measured using a
contact extensometer.
Miniaturized tensile tests were performed using a "Deben Microtest 5KN Tensile Stage".
There was no control on the stress which was applied to the miniature specimens during
fixing them in the tensile machine. For the miniaturized tests, the apparent strains (𝜀𝑎𝑝𝑝 )
were determined by dividing the recorded crosshead displacement by the initial length of
𝑟𝑒𝑓
the parallel-zone of the specimen. Furthermore, the reference elastic strain (𝜀𝑝 ) of the
miniature specimens was measured by DIC technique. The tensile tests were repeated for
the T.3 and T.4 specimens from DP and IF steels by using "Shimadzu AG-X 50KN" tensile test
13
Effects of specimen size on the tensile behavior of steels
machine, while the applied force to fix the specimens in the tensile machine was kept below
the yield strength of the specimens.
The apparent strain rates of the standard and miniaturized tests were calculated by
dividing the crosshead velocity by the initial length of the parallel-zone of the specimen. The
standard and miniaturized tensile tests were performed at apparent strain rate of 2×10 -3 s-1
at room temperature with the exception of the IF T.4 miniature specimens, which were
tested at apparent strain rate of 4×10-5 s-1. The reason of testing the IF miniature specimens
at lower strain rate was that its limited elastic elongation occurred in a few seconds. On the
other hand, the DIC method requires that a camera makes consecutive images of the
deforming specimen in a certain time interval, which was 6 seconds in the current research.
Therefore, in the case of IF miniature specimens, a lower strain rate was required to take an
adequate number of images for accurate determination of the elastic strain. Before the
tensile tests, the flat surfaces of the miniature specimens were ground using 1200 grit SiC
papers.
2.3.3 Digital image correlation
The Digital Image Correlation (DIC) method was applied for measuring the elastic strain of
the T.4 miniature specimens within the parallel-zone. DIC is an optical method that
determines the elongation of an object during the mechanical tests. With this technique a
mathematical correlation analysis is used to calculate the strain of the specimen from a
series of consecutive digital images of the specimen surface [16]. To obtain accurate results
with the DIC, the specimen needs to have a recognizable speckle pattern on its surface.
In this research, to guarantee a proper speckle pattern, the specimen surface was painted
with a white spray and then a random black pattern was finely created with a black spray. An
Oxford camera recorded images from the full parallel-zone with a resolution of 1024×768
pixels. Finally, the "digital image correlation and tracking" toolbox of the MATLAB code was
used for strain calculations. The accuracy of the DIC technique for determining strain
depends on the minimum detectable displacement, which is the spatial size of a pixel in an
image. The spatial size of a pixel can be calculated by dividing the specimen dimension to the
camera resolution [17]. In this study, the minimum displacement that can be characterized
for the T.4 miniature specimens was given by dividing the parallel length of the specimen (4
mm) by the vertical resolution of the image (1024), leading to 4×10-3 mm. Therefore, the
detection strain limit of the DIC was considered 10-3 (mm/mm) and only strains in the range
from 10-3 (mm/mm) to the yield point were determined by the DIC technique.
14
Chapter 2
Fig. ‎2-3 (a) 3D model of the one fillet of the fillet-zones with the applied boundary conditions and (b) the elastic
elongation distributions in the fillet-zone (elongation scale is in mm).
2.3.4 Finite element modelling
The proposed model for calculating the elastic elongation of one fillet of the fillet-zones
(Eq. ‎2-7) was validated by finite element simulation of the elastic elongation of a fillet-zone.
Finite element simulations were done using the commercial code ABAQUS 11.6-1. A 3D
model of one fillet of the fillet-zones of the T.4 miniature specimen was developed. The
elastic properties of the material in the simulation were taken from the measurement
performed on the M1400 standard specimens, with Young’s modulus and the Poisson’s ratio
of 210 GPa and 0.3, respectively. The fillet-zone was modelled by solid element C3D8R which
is an 8-node linear brick with reduced integration and hourglass control. As it is shown in
Fig. ‎2-3a, to simulate the tensile test, the left side of the fillet-zone was encastered while the
right side of the fillet was elongated. The fillet-zone was elongated by 0.02 mm. This is equal
to the elongation of the parallel-zone of the T.4 miniature specimen when it was deformed
by the elastic strain limit (0.005 mm/mm) of the M1400 standard specimen.
2.4
Results and discussion
Fig. ‎2-4 presents the engineering stress-strain graphs of the miniature specimens and
standard specimens for all groups of steels. The tensile curves of miniature specimens were
measured by using Deben machine. During fixing the miniature specimens to the tensile
machines, there was no control on the stress which was applied to the specimens. If the
applied force exceed the yield strength of the tested specimen, the yield strength and tensile
strength measurements would be affected. The strain of the miniature specimens was
measured based on the crosshead displacement of the tensile machine. According to this
figure, the elastic strain of miniature specimens is higher than standard specimens.
Furthermore, the yield strength and tensile strength as well as fracture strain increases by
reducing the specimen gauge length. The influence of the microtensile measurements on the
tensile behaviour of steels is discussed in this section.
2.4.1 Elastic Strain Measurement of Miniature Tensile Specimens
Results on the calculation of the elastic strain based on the proposed model and
verification of this methodology are presented and discussed in this section.
15
Effects of specimen size on the tensile behavior of steels
Fig. ‎2-4 Engineering stress-strain diagrams of miniature specimens with 3 (T.3) and 4 (T.4) mm parallel length
and standard specimens (T.120) with parallel length of 120 mm corresponding to steels (a) IF, (b) DP600, (c)
DP1000 and (d) M1400.

Experimental measurement of the elastic strain
The elastic strain-stress (𝜀ps − 𝜎) graphs of the standard and the apparent elastic strainstress (𝜀app − 𝜎) graphs of the miniature specimens for steels M1400, DP1000, DP600 and IF
are illustrated in Fig. ‎2-5. Additionally, it presents the reference elastic strain-stress (𝜀pref −
𝜎) curves of the miniature specimens that were measured by DIC. Fig. ‎2-5 shows that the
reference elastic strain-stress curves of the miniature specimens and the elastic strain-stress
of the standard specimens are in excellent agreement. This confirms that the specimen
geometry has no effect on the actual measured elastic elongation of material and the elastic
slope of the standard and reference miniaturized tests is independent of the specimen
geometry. Furthermore, the apparent elastic slope of the miniaturized tests, which were
determined from apparent elastic strain-stress (εapp − σ) graphs, are lower than the
standard ones. This indicates that the fillet-zones elongation and machine compliance
strongly increase the crosshead displacement within the miniaturized tests.

Calculation of the fillet-zones elongation and the machine compliance
To evaluate the accuracy of the developed model, the elastic elongation of one fillet (∆𝑙𝑓 )
of the fillet-zones of the T.4 miniature specimen from M1400 was simulated by finite
element analysis. The distribution of the elastic elongation in the fillet-zone is shown in
Fig. ‎2-3b and it indicates that the elastic elongation is not uniform in this zone, contrary to
the assumption of the uniform elongation in the parallel-zone and the fillet-zones which was
done by Koubaa et al. [10].
16
Chapter 2
Fig. ‎2-5 Elastic strain-stress curves of (a) M1400 (standard and T.4 miniature specimens), (b) DP1000 (standard
and T.4 miniature specimens), (c) DP1000 (standard and T.3 miniature specimens), (d) DP600 (standard and T.4
miniature specimens) and (e) IF (standard and T.4 miniature specimens) steels. The apparent elastic strain
𝑟𝑒𝑓
(𝜀𝑎𝑝𝑝 ), the reference elastic strain (𝜀𝑝 ) and the corrected elastic strain (𝜀𝑝𝑐 ) of the miniature specimens were
determined from the crosshead displacement, DIC and the proposed method, respectively. The elastic strain of
the standard specimens (𝜀𝑝𝑠 ) was determined with an extensometer.
The elongation of the fillet-zone was computed from the finite element simulation and
the proposed model (Eq. ‎2-7) and the results of both calculations are illustrated as the forceelastic elongation curves in Fig. ‎2-6. The results show that the proposed mathematical model
calculates the fillet elongation accurately.
17
Effects of specimen size on the tensile behavior of steels
Fig. ‎2-6 The elongation-force curves of one fillet of the fillet-zones of T.4 miniature specimen from steel M1400
calculated by FEM (dashed line) and Eq. ‎2-7 (dotted line).
Substituting the T.4 miniature specimens dimensions from Table ‎2-1 in Eq. ‎2-8, the
parameter 𝛼 is found to be equal to 0.19 for these specimens. Furthermore, the ratio
between the elastic elongation of one fillet of the fillet-zones and the parallel-zone was
determined for the standard specimens as 𝛼 = 0.09. The low value of 𝛼 for the standard
specimens in comparison to 𝛼 = 0.19 for the T.4 miniature specimens shows that the filletzones elongation has a much smaller effect on the crosshead displacement of the standard
specimens. For the T.4 miniature specimens from all four groups of steels, the reference
𝑟𝑒𝑓
elongation of the parallel-zone (∆𝑙𝑝 ) was measured by DIC. The machine compliance, at
𝑟𝑒𝑓
different levels of ∆𝑙𝑝 , can be determined by using Eq. ‎2-10 and subtracting the elastic
elongation of the fillet-zones and parallel-zone of the specimen from the crosshead
displacement. The machine compliance vs. applied force diagram is illustrated in Fig. ‎2-7. For
all the tested steels variations of the machine compliance versus the applied force follows
the same trend and it is independent of the material type. By interpolation of the T.4
miniature specimen from M1400 data, the compliance function of the tensile-test machine
was expressed as a bilinear curve by:
0
𝑚𝑚
8.1 × 10 (
) 𝐹 − 0.0096(𝑚𝑚)
∆𝑙𝑚 =
𝑁
𝑚𝑚
2.1 × 10−4 (
) 𝐹 − 0.0678(𝑚𝑚)
𝑁
{
−5
𝐹 < 150 𝑁
150 ≤ 𝐹 < 450 𝑁
‎2-13
450 ≤ 𝐹 < 1200 𝑁
For forces lower than 150 N, the machine compliance is insignificant and its value is
assumed zero. For all the studied steels, the contributions of the fillet-zones elongation and
the machine compliance to the crosshead displacement of one T.4 miniature specimen are
illustrated in Fig. ‎2-8. In this figure, all the specimens were deformed equivalently (2×10-3
mm) by using DIC method. In this fihure the elastic elongation of the fillet-zones and the
machine compliance were determined by using Eq. ‎2-7 and Eq. ‎2-10, respectively.
Fig. ‎2-8 shows that the elongation of the fillet-zones is equivalent for all the steels and as
it was discussed in the section ‘’Elastic elongation of the fillet-zones’’, the ratio of the
elongation of the one fillet of the fillet-zones to the elongation of the parallel-zone is
independent of the material type.
18
Chapter 2
Fig. ‎2-7 Machine compliance-applied force curve of the T.4 miniature specimens from IF, DP 600, DP1000 and
M1400. The solid line is interpolating of the M1400 miniature specimen data.
𝑟𝑒𝑓
Fig. ‎2-8 Contribution of the parallel-zone elongation (∆𝑙𝑝 ), the fillet-zones elongation (2∆𝑙𝑓 ) and machine
compliance (∆𝑙𝑚 ) on the crosshead displacement in the tensile testing of the T.4 miniature specimens. The
-3
parallel-zone elongation of all the steels was the same (2×10 mm) and the applied forces to create this
elongation were recorded.
It can be recognized that stronger specimens deform elastically up to a higher load and
thereby the machine compliance, which is function of the applied force, is larger for these
specimens. This figure also indicates that, for all the steels, the machine compliance forms
the main contribution on the crosshead displacement and its influence on the elastic strain
measurement should be precisely considered.

Validation of the proposed model
The correction procedure to calculate the elastic strain in the parallel-zone of the
miniature specimens was developed by inserting the machine compliance function (Eq. ‎2-13)
and the α value, 0.19 for the T.4 miniature specimens and 0.14 for the T.3 miniature
specimen, in Eq. ‎2-12. Then the parallel-zone strain of the miniature specimens at different
levels of the crosshead displacement and the applied force were calculated.
19
Effects of specimen size on the tensile behavior of steels
Table ‎2-2 Apparent elastic slope (𝐸 𝑎𝑝𝑝 ), reference elastic slope (𝐸 𝑟𝑒𝑓 ) and corrected elastic slope (𝐸 𝑐 ) of the
miniature specimens and elastic slope of the standard specimens (𝐸 𝑠 ). The relative error of apparent elastic
app
c
slope (𝜂 ) and corrected elastic slope (𝜂 ) were calculated based on the elastic slope of the standard
specimens. In this table, the elastic slope and relative error are given in GPa and percentage, respectively.
Specimen
T.4 M1400
T.4 DP1000
T.3 DP1000
T.4 DP600
T.4 IF
𝑬𝒔
203
212
212
202
173
𝑬𝒓𝒆𝒇
204
212
215
200
188
𝑬𝒂𝒑𝒑
55
51
50
98
88
app
𝜼
73
76
76
52
96
𝑬𝒄
204
235
220
187
165
c
𝜼
0.5
10
4
7
5
As it can be seen in Fig. ‎2-5 the corrected strain-stress (𝜀𝑝𝑐 − 𝜎) graphs of the miniature
specimens and the strain-stress (𝜀𝑝𝑠 − 𝜎) graphs of the standard specimens are in good
agreement and both miniature and standard geometries show the same elastic slope. For
each type of steel, the elastic slope of the standard specimens (𝐸 s ) was measured from
standard elastic strain-stress curves and presented in Table ‎2-2. Moreover, this table
includes the apparent elastic slope (𝐸 app ), the reference elastic slope (𝐸 ref ) and the
corrected elastic slope (𝐸 c ) of the miniature specimens which were determined from
apparent elastic strain-stress curves, reference elastic strain-stress curves and corrected
elastic strain-stress curves, respectively. The relative error of apparent elastic slope (𝜂app)
and corrected elastic slope (𝜂c) were calculated based on the elastic slope of the standard
specimens. Although the relative error of the apparent elastic strain is around 50-96% the
relative error of corrected elastic strain is less than 10%. These results show that this model
can be considered as a reliable method for calculating the elastic strain of the miniature
specimens from the crosshead displacement.
2.4.2 The influence of the specimen geometry on the yield and tensile strength
Fig. ‎2-4 shows that the yield and the tensile strength of steels increase by reducing the
parallel length. Since engineering stress-strain curves of miniature specimens were parallel
to each other in the plastic zone, the increase in the strength is due to the fact that the force
which was applied to fix the miniature specimens on the tensile test machine was not
become zero at the initial stage of the tensile. In this sense, by reducing the specimen size
(cross section area) the applied stress during the fixing the specimen results in higher stress
enhancement. This explanation was verified by tensile testing of T.4 and T.3 miniature
specimens from IF and DP 600, while the applied force during fixing the specimens was kept
very low. The results of the engineering stress-strain curves are shown in Fig. ‎2-9. As it can
be seen the yield strength and tensile strength of the specimens are independent of the
specimen gauge length.
20
Chapter 2
Fig. ‎2-9 Engineering stress-strain diagram of miniature specimens with 3 (T.3) and 4 (T.4) mm parallel length
and standard specimens (T.120) with parallel length of 120 mm corresponding to steels IF and DP600. The
applied force during the fixing the specimens on the tensile test machine was kept below the yield strength.
2.4.3 The influence of the specimen geometry on the fracture strain
According to Fig. ‎2-4 and Fig. ‎2-9, the fracture strain of the specimens increased with a
reduction of the specimen parallel length. Similar increase in the fracture strain with
decreasing the parallel length of miniature specimens have been reported by different
researchers. This can be related to the fact that post necking elongation is concentrated in
the necking region and it is independent of the specimen geometry [17]. On the other hand,
to calculate the fracture elongation, the measured deformation is divided by the initial
parallel length. Since the initial parallel length in miniature specimens is much lower than in
standard ones, the measured post necking strain in the miniature specimens is higher than in
standard specimens. In view of this problem, ISO developed the international standard ISO
2566 to eliminate the effect of the specimen parallel length on the fracture strain and enable
a better comparison of data generated from different specimen geometries. This method is
based on the Oliver formula, which is now has been widely used for conversions of fracture
strain. For specimens having the parallel length to width ratio of 4, as in the case of this
work, the Oliver equation is expressed as [18]:
0.4
√𝑆0
𝐴𝑟 = 1.74 [
]
𝐿0
𝐴,
‎2-14
where 𝐿0 and 𝑆0 are the gauge length and cross section area of the standard specimen,
respectively. 𝐴 is fracture strain of the miniature specimen and 𝐴𝑟 is the corrected strain on
gauge length L0. Eq. 2-14 has been used in [19] to correct the influence of the specimen size
on the fracture strain of different grades of steels [19].
21
Effects of specimen size on the tensile behavior of steels
Fig. ‎2-10 Comparison between fracture strains measured from miniature and standard specimens and
converted values calculated according to the Oliver’s equation.
Fig. ‎2-11 Overview of tensile strength and fracture strain of standard and miniature specimens. In the
microtensile tests, the applied force during fixing specimens is kept below the yield strength.
Substituting the dimensions of standard specimen from Table ‎2-1 into Eq. ‎2-14 the
following equation is obtained for correcting the measured fracture strain of miniature
specimens:
𝐴𝑟 = 0.55𝐴.
‎2-15
The fracture strain of the miniature specimens with 3 mm and 4 mm gauge length were
corrected and compared with the values obtained from standard specimens in Fig. ‎2-10. The
corrected fracture strains of DP600 and DP1000 steels are in good agreement with the
fracture strains of the standard specimens. For the M1400 miniature specimens, the
corrected fracture strain is 9% while the fractures strain of the standard specimens was 5%.
The difference between the corrected fracture and the strain of the standard specimens can
be attributed to effect of the machine compliance on the crosshead displacement.
Considering that the machine compliance is a function of the applied force the influence of
the machine compliance on the strain is more significant in case of materials with high
strength like M1400 than in material like DP600 and DP1000. The corrected fracture strains
of IF miniature specimens is lower than the standard ones. This was expected, since the
international standard ISO 2566 explained here can be only applied to low carbon steels
while IF steel is carbon free.
22
Chapter 2
To illustrate the effect of the specimen geometry on the tensile properties, the fracture
strain-tensile strength curves of the standard and miniature specimens are shown in
Fig. ‎2-11. The tensile behavior of the steels is not affected by specimen dimensions and the
corrected fracture strain of miniature specimens gives an estimation of failure strain of the
standard specimens. Therefore, the miniature specimens can be used to determine the
tensile behaviour of steels. However, it is required to correct the influence of the machine
compliance on the measured elastic strain, if the strain is measured from the crosshead
displacement.
2.5
Conclusions
The influence of the specimen geometry on the tensile behaviour of different types of
steels (M1400, DP1000, DP600 and IF) is studied and the results are summarised as
following:

The specimen geometry has insignificant influence on the actual measured elastic
strain of the materials. Measurements recorded with the crosshead displacements
on the miniature specimens displayed higher strain as a result of the effect of the
elastic strain of the fillet-zones and the machine compliance.

A mathematical model is proposed to calculate the elastic strain of the fillet-zones
and the machine compliance. The mathematical model is experimentally
evaluated for miniature specimens from different types of steels and different
dimensions. For each type of steel, the calculated elastic strain and the strain
measured on the standard specimens are in excellent agreement and
consequently the proposed model can be used for calculating the elastic strain of
the miniature specimens from the crosshead displacement.

The yield strength and tensile strength does not change by reducing the parallel
length from 120 mm to 3 mm.

The fracture strain of the miniature specimens is higher than the standards values.
This is a result of the calculation method, since the fracture strain is calculated by
dividing the elongation by the initial gauge length, which is smaller for miniature
specimens. Since the post-uniform deformation is independent of the specimen
parallel zone, the measured fracture strain is higher in miniature specimens. The
Oliver equation was applied for correcting the fracture strain of the miniature
specimen to the strain obtained from standard ones. It was found that this
equation accurately corrected the fracture strain of the low carbon steels, but
failed for the case of martensitic and interstitial free steels.
23
Effects of specimen size on the tensile behavior of steels
2.6
References
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24
Chapter 2
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25
Effects of specimen size on the tensile behavior of steels
26
CHAPTER 3
3 An Improved X-ray Diffraction Analysis
Method to Characterize Dislocation
Density in Lath Martensitic Structures*
Abstract
An improved X-ray diffraction line profile analysis method is developed to determine
dislocation density of lath martensitic steels. This method combines the modified WarrenAverbach (MWA) and the modified Williamson-Hall (MWH) methods. The developed method
is stable under different initial conditions and leads to unique values for the dislocation
density, the effective outer cut-off radius of the dislocations (Re) and the dislocations
distribution parameter (M). Dislocation structure of lath martensite in a steel, in the asquenched as well as tempered conditions, are characterized by using the proposed method.
The calculated dislocation density is compared with the values obtained from the MWH
method by considering a constant value for M. It was found that both methods provide
dislocation densities in the range of the values calculated from the dislocation strengthening
component of the yield strength.
Keywords: Dislocations, X-ray diffraction, Martensite, Mechanical characterization
*
This chapter is based on a scientific paper:
F. HajyAkbary, J. Sietsma, A. J. Bӧttger and M. J. Santofimia, An Improved X-ray Diffraction Analysis Method to
Characterize Dislocation Density in Lath Martensitic Structures, Material Science and Engineering A, vol. 639,
pp. 208-218, 2015.
27
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
3.1
Introduction
Knowledge of the microstructural properties and their effects on the yield strength is
required to tailor the mechanical properties of steels. Dislocation density as a crucial factor
influencing the yield strength can be determined by applying Transmission Electron
Microscopy (TEM) and X-ray diffraction (XRD) line profile analysis [1, 2]. It should be kept in
mind that in an inhomogeneous system such as martensitic steel, where the dislocation
density varies from place to place within a grain on a sub-micron scale, the XRD method
gives a macroscopic average value while TEM gives a microscopic local value [3].
Furthermore, in the case of martensitic microstructures with a high density of dislocation
(higher than 1014 m-2) applying TEM is difficult. This is because of the complicated image
contrasts from the sample [2]. Therefore, especially at high dislocation densities, like in
highly deformed metals or in martensitic steels, XRD offers a promising alternative.
However, the quantification of the dislocation density from an XRD pattern of broadened
peaks is not straightforward.
The modified Williamson-Hall (MWH) method [4] is known as an accessible method in
XRD line profile analysis. This method determines the dislocation density if the dislocations
distribution parameter (M) is known. It is argued that M depends on the effective outer cutoff radius of the dislocations (𝑅𝑒 ) and the dislocation density (𝜌) [2]. No direct method has
been used to determine M and it can only be obtained from 𝑀 = 𝑅𝑒 √𝜌 [5] in which 𝑅𝑒 is
calculated from the MWA method. This means that the MWH equation includes two
unknown parameters, 𝜌 and 𝑀. Therefore, the MWH approach has been applied under the
assumption of a fixed value for M, as a qualitative method in limited number of research [6,
7].
An alternative method for XRD line profile analysis is the modified Warren-Averbach
(MWA) which has been widely used to determine the dislocation density and the effective
outer cut off radius of the dislocations [2, 3, 8, 9]. Generally, this method is used by
assuming the strain function of the dislocations (the Wilkens function) as a logarithmic
function of (𝑅𝑒 /𝐿) [10] where 𝐿 is the Fourier length. The strain function
phenomenologically describes the dislocation-dislocation correlations that appear in high
order Fourier coefficients [11]. More details about 𝐿 and strain function are given in
section ‎3.2.2. This approach was applied by Movaghar et al. [9] to study the influence of
severe plastic deformation on the dislocation structure of a martensitic steel. Furthermore,
the same approach was used to evaluate the dislocation density of a 11Cr-0.1C (wt.%)
martensitic steel after annealing [12]. It should be recognized that calculation of the
dislocation density by assuming a logarithmic strain function is applicable only at small L
values (𝐿 < 2.88 𝑅𝑒 ) [5]. Although this method has been used widely, it is not a robust
method. The reason is that 𝑅𝑒 is unknown and it is not possible to determine the relevant
range of L. Additionally, this approach depending on the assumed range of L gives different
𝑅𝑒 and dislocation density. An alternative approach in the MWA method is defining the
strain function for whole ranges of L, e. g. 𝐿 < 2.8816 𝑅𝑒 as well as 𝐿 > 2.8816 𝑅𝑒 [10].
Although this approach is valid for any L, it is not robust also and the results depend on the
initial conditions and the assumed range of L. In conclusion, for the determination of the
dislocation density, none of the XRD line profile analysis methods has been generally
accepted.
28
Chapter 3
A quantitative value for the dislocation density can also be calculated from the dislocation
strengthening component of the yield strength. The dislocation strengthening component is
determined by subtracting the contributions of the other strengthening components,
including lattice friction stress, the solid solution strengthening, the grain boundary
strengthening and the precipitation strengthening, from the total yield strength.
Subsequently, the dislocation density can be estimated from the relation between
dislocation strengthening and dislocation density. Since the yield strength is a bulk property,
the obtained dislocation density value represents the bulk dislocation density. In this sense,
the calculated dislocation density can be used to validate the dislocation density determined
by using the X-ray diffraction analysis methods.
In the present chapter, an improved approach has been developed for XRD line profile
analysis by combining the MWH and MWA methods. This approach provides an expression
for the Fourier coefficients that is valid for any range of L and is stable under different initial
conditions. The developed method is applied to determine the dislocation density, the
effective outer cut-off radius of the dislocations (Re) and the dislocations distribution
parameter (M) of lath martensite in a steel under the as-quenched as well as tempered
conditions. The calculated dislocation densities are compared with the values obtained from
the MWH method by considering a constant value for M. It will be shown that the calculated
dislocation densities from XRD analysis methods are in a good agreement with the values
that are obtained from contribution of the dislocation strengthening to the yield strength.
3.2
Calculation of dislocation density
3.2.1 Analysis of XRD peak broadening by Modified Williamson-Hall Method
The XRD peak broadening caused by strain has long been used to characterize dislocation
density. For isotropic materials, the Williamson-Hall equation approximates the dislocation
density from X-ray peak broadening as [13]:
∆𝐾 ≅
𝛼𝑠
𝐷
+𝑁𝑏√𝜌 𝐾,
‎3-1
where ∆𝐾 is the peak width, 𝑁 is a constant (0.263), 𝛼𝑠 is the shape factor, 𝐷 is the
crystallite size, 𝐾 is the magnitude of the diffraction vector, 𝑏 is the magnitude of the
Burgers vector and 𝜌 is the dislocation density. Here, 𝛼𝑠 is given 0.9 under the assumption of
spherical crystals with cubic symmetry [14] (thereafter 𝛼𝑠 is assumed 0.9) and 𝐾 is obtained
by 𝐾 = 2𝑠𝑖𝑛𝜃/𝜆, in which 𝜃 and 𝜆 are the diffraction angle and the wavelength,
respectively. In principle this equation is valid for each {hkl} reflection and the dislocation
density is obtained by fitting Eq. ‎3-1 to a plot of ∆𝐾 versus 𝐾. However, in cases of strong
strain anisotropy, such as observed in lath martensitic steel, ∆𝐾 is not a linear function of 𝐾
[2]. In these applications, this method overestimates the dislocation density [3].
Ungar et al. [4] developed a modified Williamson-Hall (MWH) method by accounting the
influence of the strain anisotropy. To do this, they defined a scaling parameter, 𝐶̅ , which is
called the average contrast factor of dislocations. The MWH equation is written as [6]:
∆𝐾 ≅
0.9
𝐷
𝜋
+𝑏𝑀√ 2 𝜌 (𝐾𝐶̅ 1/2 ).
‎3-2
29
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Deviations from this approximation are of the order of 𝐾 2 𝐶̅ . The dislocation contrast
factor, 𝐶̅ , is a function of the Miller indices and can be determined by applying an approach
that is given in the following paragraph. M is a dimensionless constant and it is known as the
dislocations distribution parameter. As it was mentioned in section ‎3.1, no direct method is
available to determine 𝑀 and it can only be obtained on the basis of the relation 𝑀 = 𝑅𝑒 √𝜌,
where 𝑅𝑒 is determined by applying the MWA method [15]. Accordingly, the intercept and
the slope of the MWH plot, ∆𝐾 versus 𝐾𝐶̅ 1/2 , give D and the product 𝑀√𝜌, respectively.
Here, 𝑀√𝜌 is denoted as 𝛾 ( = 𝑀√𝜌) and the dislocation density can be calculated given a
value of 𝑀.
The {hkl} dependence of the dislocation contrast factor (𝐶̅ ) in an untextured cubic
polycrystalline material is given by [16]:
𝐶̅ = ̅̅̅̅̅̅
𝐶ℎ00 (1 − 𝑞𝐻 2 ),
‎3-3
where 𝑞 is a parameter that depends on the edge or screw character of the dislocations and
ℎ2 𝑙2 +ℎ2 𝑘 2 +𝑙2 𝑘 2
̅̅̅̅̅̅
will be determined experimentally and 𝐻 2 =
2
2 2 2 . 𝐶ℎ00 is the average dislocation
(ℎ +𝑘 +𝑙 )
contrast factor for the {ℎ00} reflections and is determined by the dislocation contrast factor
for the {ℎ00} reflections (𝐶ℎ00 ) of pure screw and edge dislocations as well as the fractions
of screw and edge dislocations. For pure edge and screw dislocations, 𝐶ℎ00 is determined by
the elastic parameters of the material (𝐶11 , 𝐶12 and 𝐶44 ) and using [17]:
𝑐
𝐶ℎ00𝑖 = 𝑎𝑖 ℎ00 [1 − 𝑒𝑥𝑝 (
𝐴
𝑐
𝐶
𝑏𝑖 ℎ00
𝑐
)] + 𝑐𝑖 ℎ00 𝐴 + 𝑑𝑖 ℎ00 ,
‎3-4
here 𝐴 is the elastic anisotropy parameter and is given as 𝐴 = 𝐶
𝑐
𝐶
𝑐
2𝐶44
11 −𝐶12
𝑐
. The parameters
𝑎𝑖 ℎ00 , 𝑏𝑖 ℎ00 ,𝑐𝑖 ℎ00 and 𝑑𝑖 ℎ00 (𝑖= edge or screw) are determined by the elastic constants of the
material [17]. Then the average contrast factor of the {ℎ00} reflections, ̅̅̅̅̅̅
𝐶ℎ00 , can be
estimated by considering the fractions of edge and screw dislocations. In the following a
method is given to calculate fractions of screw and edge dislocations. Combining Eq. ‎3-2 and
Eq. ‎3-3 and substituting ̅̅̅̅̅̅
𝐶ℎ00 , the value of 𝑞 can be obtained from the following equation:
(∆𝐾−𝛼)2
𝐾2
≅ 𝛽 2 ̅̅̅̅̅̅
𝐶ℎ00 (1 − 𝑞𝐻 2 ),
where 𝛼 =
0.9
𝐷
‎3-5
𝜋𝜌
and 𝛽 = 𝑏𝑀√ 2 . The experimental value of 𝛼 is determined by imposing a
linear relationship between the left-hand term and 𝐻 2 . Then, the inverse value of 𝑞 is given
by the intercept of the extrapolated line with the horizontal axis. Finally, 𝐶̅ is obtained for
each reflection by replacing the 𝑞 value in Eq. ‎3-3. Knowing the experimental value of 𝑞, the
fraction of screw or edge dislocations can be given [2]:
𝑓 𝑒𝑑𝑔𝑒 =
𝑡ℎ
𝑞𝑠𝑐𝑟𝑒𝑤
−𝑞
𝑡ℎ
𝑡ℎ
𝑞𝑠𝑐𝑟𝑒𝑤
−𝑞𝑒𝑑𝑔𝑒
= 1 − 𝑓 𝑠𝑐𝑟𝑒𝑤 ,
‎3-6
where 𝑓 𝑒𝑑𝑔𝑒 and 𝑓 𝑠𝑐𝑟𝑒𝑤 are the fractions of edge and screw dislocations, respectively.
Here, 𝑞𝑖𝑡ℎ (𝑖= edge or screw) is the theoretical value of 𝑞 for the pure edge or screw
dislocations and can be calculated from the following equation [17]:
30
Chapter 3
𝐴
𝑞𝑖𝑡ℎ = 𝑎𝑖𝑞 [1 − 𝑒𝑥𝑝 (𝑏𝑞)] + 𝑐𝑖𝑞 𝐴 + 𝑑𝑖𝑞 ,
‎3-7
𝑖
in which the parameters of 𝑎𝑖𝑞 , 𝑏𝑖𝑞 , 𝑐𝑖𝑞 and 𝑑𝑖𝑞 depend on the elastic constants of the
material [17].
3.2.2 Analysis of XRD peak broadening by combining Modified Williamson-Hall and
Modified Warren-Averbach methods
In the modified Warren-Averbach (MWA) method, the experimentally measured
diffraction pattern is expressed as a Fourier series [4]. The real part of the Fourier
coefficients, 𝐴(𝐿), is considered as the summation of a size Fourier coefficient (𝐴𝑠 ) and a
strain Fourier coefficient. The modified Warren–Averbach equation is expressed as [4]:
𝑙𝑛 𝐴(𝐿) ≅ 𝑙𝑛 𝐴𝑠 (𝐿) − 𝜌
𝜋𝑏 2 2
𝐿 𝑓(𝜂)𝐾 2 𝐶̅ .
2
‎3-8
Deviations from this approximation are of the order of 𝐾 4 𝐶̅ 2 . 𝐿 is given as: 𝐿 = 𝑛𝑎3 ,
where 𝑛 represents the integers starting from zero and 𝑎3 = 2(sin𝜃
𝜆
2 −sin𝜃1 )
in which
(𝜃2 − 𝜃1 ) is the angular range of the measured diffraction profile. Here, each domain is
represented by columns of cells along the 𝑎3 direction and 𝐿 indicates the undistorted
distance between a pair of cells along the 𝑎3 direction [18, 19]. The dislocations contrast
factor, 𝐶̅ , is calculated based on the method described in section ‎3.2.1. Furthermore, an
expression to determine 𝐴(𝐿) from the XRD peak parameters is given in section ‎3.3.3. In
Eq. ‎3-8, 𝑓(𝜂) is the strain function (the Wilkens function) which phenomenologically
describes the dislocation-dislocation correlations that appear in high order Fourier
coefficients [11] and it has the following explicit form [19]:
𝑓(𝜂)
𝜂
7
512
2
1
arcsin𝑉
−ln(𝜂) + ( − ln 2) +
+ (1 − 2 ) ∫
d𝑉
𝑖𝑓 𝜂 ≤ 1
4
90𝜋𝜂 𝜋
4𝜂
𝑉
0
1
1 769 41𝜂 2𝜂3
1 11
7 𝜂3
𝜂2
− (
+
+
) (1 − 𝜂2 )2 − (
+
+
)
arcsin𝜂
+
𝜋 180𝜂 90
90
𝜋 12𝜂2 2 3
6
=
512
11 1
1
− ( + ln 2𝜂) 2
{ 90𝜋𝜂
24 4
𝜂
‎3-9
𝑖𝑓 𝜂 ≥ 1
𝐿
where 𝜂 = 2.8816(𝑅𝑒). Generally, only the first part of the strain function (𝑓(𝜂) for 𝜂 ≤ 1) is
𝑅
considered and it is simplified as 𝑓(𝜂) = 𝐿𝑛 ( 𝐿𝑒) [2, 9]. This approach has three drawbacks;
(a) only a limited part of the Fourier function is used and therefore does not include the real
character of the dislocations and (b) 𝑅𝑒 is unknown and it is not possible to determine the
relevant range of L, (c) depending on the assumed range of L, different values for 𝑅𝑒 and
dislocation density are obtained. Thus, it is more convenient to apply the strain function for
the whole range of 𝐿 (Eq. ‎3-9), i. e. 𝐿 < 2.8816 𝑅𝑒 as well as 𝐿 > 2.8816 𝑅𝑒 [10]. The
31
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
dislocation density and the effective outer cut-off radius of the dislocations can be
determined from the second term in Eq. 3-8 by defining the parameter X(L) as:
𝑋(𝐿) = 𝜌
𝜋𝑏 2
2
𝐿2 𝑓(𝜂),
‎3-10
in which X(L) is derived as a function of L from the slope of 𝑙𝑛𝐴(𝐿) vs. 𝐾 2 𝐶̅ plot (Eq. ‎3-8).
Subsequently, Eq. ‎3-10 can be rewritten:
𝑋(𝐿)
𝐿2
=𝜌
𝜋𝑏 2
2
𝑓(𝜂).
‎3-11
Plotting of the left hand side of Eq. ‎3-11 vs. 𝐿 gives the dislocation density (𝜌) and the
effective outer cut-off radius of dislocation (𝑅𝑒 ). It has been shown that this approach is not
robust and results in different values for ρ and 𝑅𝑒 depending on the initial conditions and
the assumed range of L [10]. In this research, in order to calculate values of 𝜌 and 𝑅𝑒
independently from the range of L, the MWA and MWH methods are combined. This is done
by considering 𝑀 = 𝑅𝑒 √𝜌 [18] and =𝑀√𝜌. It is worth to remind that in the section ‎3.2.1, 𝛾
is evaluated by fitting Eq. ‎3-2 to the MWH plot. Then Eq. ‎3-11 can be rewritten as:
𝑋(𝐿)
𝐿2
=𝜌
𝜋𝑏 2
2
𝑓(𝜂∗ ),
in which 𝜂 = 2.8816
‎3-12
𝐿𝜌
𝛾
and 𝑓(𝜂) is given by ‎3-9. Fitting
𝑋(𝐿)
𝐿2
versus L gives the dislocation
density as the fitting parameter. The developed approach is based on the fact that the MWH
and MWA methods both give similar values for the dislocation density as has been shown in
[15] by Ungar et al.
3.2.3 Determination of dislocation density from dislocation strengthening
Generally, a linear contribution of the strengthening mechanisms is considered to the
yield strength of lath martensite [20]. However, the linear approximation overestimates the
yield strength [21, 22]. Alternatively, the following approach has been considered for the
expression of the yield strength (𝜎𝑦 ) of lath martensite [22]:
1/2
2
𝜎𝑦 = 𝜎0 + 𝜎𝑠𝑠 + 𝜎𝑔𝑏 +(𝜎𝜌2 + 𝜎𝑝𝑐𝑝𝑡
)
,
‎3-13
where 𝜎0 is the lattice friction stress for pure Fe, 𝜎𝑠𝑠 is the solid solution strengthening, 𝜎𝑔𝑏
is the grain boundary strengthening, 𝜎𝜌 is the dislocation strengthening and 𝜎𝑝𝑐𝑝𝑡 is the
precipitation strengthening. Then by subtracting the contributions of 𝜎0 , 𝜎𝑠𝑠 , 𝜎𝑔𝑏 , 𝜎𝑝𝑐𝑝𝑡
from the total yield strength, the contribution of the dislocation strengthening can be
determined. Finally, the dislocation density can be obtained from the relation between
dislocation density and dislocation strengthening. More details on the calculation procedure
of the strengthening mechanisms are given in the following.
Solid solution strengthening. The solid solution strengthening (𝜎𝑠𝑠 ) is caused by solid
solution strengthening of interstitial carbon atoms and substitutional atoms as:
𝜎𝑠𝑠 = 𝜎𝐶 + 𝜎𝑠𝑡 ,
32
‎3-14
Chapter 3
in which 𝜎𝐶 is the solid solution strengthening of carbon atoms and 𝜎𝑠𝑡 is the solid solution
from substitutional atoms. The solid solution strengthening from carbon can be calculated
by [23, 24]:
1/3
𝜎𝐶 = 1171.3 𝑋𝐶 𝑒𝑥 𝑝 [−4.07 × 104 𝑡 𝑛 (𝑋𝐶𝑎𝑡 )0.635 𝑒𝑥𝑝 (−
33598
𝑅𝑇
)],
‎3-15
where 𝑋𝐶 is the average concentration of carbon in solid solution in wt.%, 𝑡 is time in hour, n
is a rate constant as 0.62, 𝑋𝐶𝑎𝑡 is the atom fraction of carbon, 𝑅 is the universal gas constant
(8.31441 𝐽𝑚𝑜𝑙 −1 𝐾 −1) and 𝑇 is the absolute temperature of the tempering process. The
exponential term in Eq. ‎3-15 considers the effect of tempering process on the carbon solid
solution strengthening. The solid solution strengthening from substitutional atoms is
expressed as [25]:
𝑎𝑡 0.75
𝑎𝑡 0.75
𝑎𝑡 0.75
𝑎𝑡 0.75 ],
𝜎𝑠𝑡 = 0.689 [110(𝑋𝑆𝑖
)
+ 70(𝑋𝑀𝑛
)
+ 61(𝑋𝑁𝑖
)
+ 14(𝑋𝐶𝑟
)
‎3-16
where 𝑋𝑖𝑎𝑡 is the atomic concentrations of element i (=Si, Mn and etc).
Grain boundary strengthening. In order to determine the grain boundary strengthening,
a proper definition of the concept ''grain'' for martensite is needed. It is believed that highangle boundaries arrest dislocation motion, while dislocations can propagate across lowangle boundaries, like the lath boundaries that have a misorientation angle of 2.8–2.9° [20].
Moreover, it has been shown in [20] that the majority of high angle boundaries in the
martensitic structure are block boundaries, and a minority are packet boundaries. Therefore,
a more important effect is expected for the block boundaries than for the packet
boundaries. The dominant influence of the block boundaries on the strengthening of
martensite structures has been reported in [20, 26, 27] and therefore in the present
research the block boundaries are considered as the effective grain boundary. The grain
boundary strengthening (𝜎𝑔𝑏 ), in MPa, based on the Hall-Petch equation is calculated as
[26]:
𝜎𝑔𝑏 =
𝑘𝐻𝑃
√𝑑𝑏
,
‎3-17
in which 𝑑𝑏 is the martensite block size and 𝑘HP is Hall-Petch slope and is given as 0.21
MPa.m½ by Shibata et al. for block boundaries [26].
𝜀
Precipitation strengthening. The precipitation strengthening due to the ε-carbide, 𝜎𝑝𝑐𝑝𝑡
(MPa), is expressed by [28]:
𝜀
𝜎𝑝𝑐𝑝𝑡
=(
13320√𝑓𝜖
𝑑𝜀 (√𝜋−2√𝑓𝜖
𝑑
𝜀
)𝑙𝑛(0.496
),
)
‎3-18
in which 𝑑𝜀 (nm) is the diameter of the 𝜀-carbide and 𝑓𝜀 is the volume fraction of 𝜀-carbide.
𝜃
For the cementite precipitates, the precipitation strengthening (𝜎𝑝𝑐𝑝𝑡
), in MPa, is given by
the Ashby-Orowan equation [29]:
𝜃
𝜎𝑝𝑐𝑝𝑡
=(
12223√𝑓𝜃
𝑑𝜃
𝑑
𝜃
) 𝑙𝑛 (0.568
),
‎3-19
33
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
where 𝑑𝜃 (nm) is the diameter of the cementite particles and 𝑓𝜃 is volume fraction of
𝜀
𝜃
cementite. In presence of both carbides in the microstructure, the sum of 𝜎𝑝𝑐𝑝𝑡
and 𝜎𝑝𝑐𝑝𝑡
is
accounted as precipitation strengthening.
Dislocation strengthening. The contribution of the dislocations on the yield strength, 𝜎𝜌 ,
can be approximated by substituting numerical values of 𝜎𝑦 , 𝜎0 , 𝜎𝑠𝑠 , 𝜎𝑔𝑏 , 𝜎𝑝𝑐𝑝𝑡 into Eq. ‎3-13.
Then, the dislocation density, 𝜌 (m−2 ), is obtained from the well-known Taylor equation:
𝜌 = (𝛼
𝜎𝜌
𝑑 𝑀𝑑
2
) ,
𝐺𝑏
‎3-20
where 𝑀𝑑 is the Taylor factor, 𝛼𝑑 is a geometrical constant and 𝐺 is the shear modulus of
the material.
3.3
Experimental procedure
3.3.1 Material and treatments
In the present work, a steel with nominal chemical composition of 0.3C-1.6Si-3.5Mn
(wt.%) (1.3C-2.9Si-3.5Mn (at.%)) is studied. The 3.5 mm thick steel sheets were received in
the hot-rolled condition and the microstructure consisted of a martensite fraction of 0.91
and a retained austenite fraction of 0.09. Cylindrical specimens with a length of 10 mm and a
diameter of 3.5 mm were machined parallel to the rolling direction of the sheets. To obtain a
fully lath martensitic structure, the specimens were austenitized at 1173 K for 180 s and
then quenched to room temperature at a rate of -20 K.s-1. The treatments were followed by
tempering the specimens at 673 K for 5 s, 10 s, 50 s, 100 s and 200 s and finally quenching
them to room temperature. The heat treatments were performed in a Bähr DIL 805 A/D
dilatometer. In this paper, the QT-y code identifies the specimen that was quenched and
subsequently tempered at 673 K for y seconds and the QT-0 code refers to the as-quenched
specimen.
3.3.2 Microstructure observation
The QT-0 and QT-200 specimens were metallographically prepared for EBSD examination
with a final polishing step of 0.05 μm using an OPS suspension for 20 minutes. The
specimens were analysed by orientation imaging microscopy (OIM) on a FEI Nova 600
Nanolab dual-beam (focused ion beam) scanning electron microscope equipped with a FEG
column. The analysis was performed under the following conditions: acceleration voltage 20
kV; working distance 25 mm; tilt angle 70°; step size 50 nm. The orientation data were postprocessed with the TSL system. To avoid artificial influence of the clean-up procedure on the
grain size measurement, instead of cleaning a threshold value for the confidence index was
considered. For calculating the grain size distribution only data with a confidence index
greater than 0.2 were analysed. The combined Image Quality (IQ) and Inverse Pole Figure
(IPF) maps were corrected by applying the grain dilatation procedure provided by TSL
software. The SEM study was made on the electropolished cross-sections after etching with
2% Nital, using a JEOL JSM-6500F field emission gun scanning electron microscope (FEGSEM) operating at 15 kV.
34
Chapter 3
3.3.3 X-ray diffraction measurement
Specimens for X-ray diffraction (XRD) analysis were polished mechanically and
electrolytically. Electrolytic polishing was done with a solution of 78 ml perchloric acid, 90 ml
distilled water, 730 ml ethanol and 100 ml 2-butoxyethanol at 40 V for 15 s. X-ray diffraction
experiments were performed using a Bruker type D8-Advance diffractometer in BraggBrentano geometry with graphite monochromator equipped with a Bruker Vantec Position
Sensitive Detector (PSD) using a Cu anode. Measurements were performed in the 2ϴ range
of 40° (2θ) to 150° (2ϴ) with a step size of 0.02° (2ϴ) and a counting time of 10 s. The
reflections of {110}, {200}, {211}, {220}, {310} and {222} of the BCC structure were measured.
The base line of the XRD profile was removed. Then, K α2 elimination was performed and the
intensity was corrected with the Lorentz Polarization factor [1]. The modified WilliamsonHall (Eq. ‎3-2) and the modified Warren-Averbach (Eq. 8) methods are based on the
diffraction vector, 𝐾. In this matter, the diffraction intensity was plotted versus 𝐾 using
2sin𝜃
𝐾 = 𝜆 . In the case of CuKα1 radiation, 𝜆 = 0.15405 nm. Then, the integrated intensity of
each reflection was normalized by [12]:
+∞
∫−∞ 𝐼(𝜃)𝑑𝜃 = 1,
‎3-21
in which 𝐼(𝜃) is the X-ray diffraction intensity. The diffraction peaks can be fitted well with a
Voigt function, which is a convolution of the Lorentzian and the Gaussian functions. The
fitting was done by using the Peak Analyser toolbox of Origin 9.0 programme which provides
L
the position, full-width at half-maximum (FWHM) of the Lorentzian function (Wm
) and
G
FWHM of the Gaussian function (Wm ). The correction of the instrumental broadening was
done based on the LaB6 SRM660a [31] and using the following equations [32]:
(𝑊𝐶𝐺 )2 = (𝑊𝑚𝐺 )2 − (𝑊𝑠𝐺 )2 ,
‎3-22
𝑊𝐶𝐿 = 𝑊𝑚𝐿 − 𝑊𝑠𝐿 ,
‎3-23
where the subscripts 𝑐, 𝑚 and 𝑠 refer to the corrected, the measured and the standard
specimen, respectively. Finally, the FWHM of the diffraction peaks (Δ𝐾) was given by [33]:
1
𝛥𝐾 = 2 {1.0692𝑊𝐶𝐿 + √0.86639(𝑊𝐶𝐿 )2 + 4(𝑊𝐶𝐺 )2 }.
‎3-24
Furthermore, the real part of the Fourier coefficient was calculated by replacing 𝑊𝐶𝐿 and
𝑊𝐶𝐺 in the following expression:
1
𝐴(𝐿) = 𝑒𝑥𝑝(−(2 𝐿2 (𝑊𝐶𝐺 )2 + 𝐿𝑊𝐶𝐿 )),
‎3-25
3.3.4 Microtensile test
Two dog bone specimens were prepared in the axial direction of each dilatometry
cylinder. The specimens were made with an electron discharge machine (EDM) establishing a
gauge length of 3 mm and a cross section of 0.8 × 0.8 mm2. Before the tensile test, the flat
surfaces of the miniature specimens were ground using 1200 grit SiC paper.
35
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Fig. ‎3-1 SEM micrograph of (a and b) the specimen QT-0 (as-quenched), (c and d) the specimen QT-5 and (e and
f) the specimen QT-200. For each specimen, the first and the second images show the micrograph at low and
high magnifications, respectively.
Microtensile tests were performed using a Deben Microtest 5000N Tensile Stage.
Elongation of the miniature specimens was determined by subtracting the machine
compliance from the total crosshead displacement, on the basis of the approach that was
given in chapter 2 ([34]). This step is taken due to the fact that the machine compliance is
recorded along with the specimen elongation as the crosshead displacement.
3.4
Results and discussion
3.4.1 Microstructure characterization

Carbide precipitation
Fig. ‎3-1a shows a SEM micrograph of the specimen QT-0 (as-quenched) at low
magnification. Due to the fine size of the carbides, it is rather difficult to identify carbide
formation in this figure. However, at higher magnification in Fig. ‎3-1b, a noticeable density of
carbides is detected. Carbide formation during quenching of the specimen QT-0 is related to
the auto-tempering of martensite. Auto-tempering means that tempering started during the
quenching process, i.e. carbon diffusion resulting in segregation of carbon and carbide
precipitation. Formation of ε-carbide during quenching of low and medium carbon steels has
been often reported [25, 40] and therefore the formed carbide in the present specimen is
considered as ε-carbide. Due to the high fraction of carbides, here it is assumed that all
carbon present in the martensitic structure forms 𝜀-carbide during the quenching.
Accordingly, the volume fraction of 𝜀-carbide (Fe3C) in the QT-0 specimen is approximated as
0.05. This is an acceptable value according to the SEM image of the specimen QT-0
(‎Fig. ‎3-1b). It is important to emphasize that under this assumption, the amount of carbon in
solid solution strengthening carbon in the specimen QT-0 is zero.
36
Chapter 3
Fig. ‎3-2 Simulation of cementite formation during tempering at 673 K, based on ‎3-26 and ‎3-27.
Fig. ‎3-1c and 1d show SEM images of the specimen QT-5 at low and high magnifications,
respectively. Moreover, SEM micrograph of the specimen QT-200 at low magnification is
presented in ‎Fig. ‎3-1e and at high magnification is shown in ‎Fig. ‎3-1f. Both specimens consist
of etched, tempered and carbide-filled regions. The distribution of carbide morphology in
the specimens QT-5 and QT-200 can be detected at high magnification images in Fig. ‎3-1d
and 1f, respectively.
ThermoCalc simulations showed that cementite is stable at the tempering temperature
(673 K). Kinetics of cementite formation is examined based on the Johnson-Mehl-Avrami
(JMA) equation applied in [24] for Si added steels as:
𝑓𝜃 (𝑡) = 1 − 𝑒𝑥𝑝{−𝑘𝐽𝑀𝐴 𝑡 𝑛 },
‎3-26
in which 𝑓𝜃 (𝑡) is the volume fraction of cementite normalised by its equilibrium volume
fraction at the tempering temperature and KJMA and n are rate constants. Here, n is
considered 0.62 and 𝑘𝐽𝑀𝐴 , in hour −n , is assumed to obey the relation developed by
Bhadeshia et al. for 0.43C-2.0Si-3.0Mn (wt.%) steel [24]:
𝑘𝐽𝑀𝐴 = 550 𝑒𝑥𝑝{−(33589 𝐽. 𝑚𝑜𝑙 −1 )/𝑅𝑇},
‎3-27
The volume fraction of equilibrium cementite at the tempering temperature (673 K) is
determined as 0.05 from ThermoCalc. Fig. ‎3-2 illustrates the kinetics of cementite formation
during tempering according to Eq. ‎3-26 and Eq. ‎3-27. While during the first 50 s of the
tempering process there is a negligible cementite formation, longer tempering process
increases the rate of cementite formation and finally after 200 s of tempering the fraction of
cementite reaches to 0.01. Thereby, it is assumed that during 50 s of the tempering fine 𝜀carbide particles start to dissolve and provide carbon as solid solution in martensite. In
longer tempering time, carbon in solid solution nucleates as cementite and big 𝜀-carbide
particles dissolve at the expense of the growth of cementite. This is consistent with general
understanding from transition of 𝜀-carbides to cementite in which 𝜀-carbides decomposes
prior to cementite formation [35].
37
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Table ‎3-1 Amount of carbon in solid solution (𝑋𝐶 ), fraction of 𝜀-carbide (𝑓𝜀 ), particle size of 𝜀-carbide (𝑑𝜀 ),
fraction of cementite (𝑓𝜃 ) and particle size of cementite (𝑑𝜃 ).
Specimen
QT-0
QT-5
QT-10
QT-50
QT-100
QT-200
𝑿𝑪 (wt.%)
0
0.05±0.05
0.05±0.05
0.05±0.05
0.05±0.05
0.05±0.05
𝒇𝜺
0.05±0.005
0.045±0.005
0.045±0.005
0.040±0.005
0.035±0.005
0.030±0.005
𝒅𝜺 (𝒏𝒎)
5±5
10±5
10±5
10±5
10±5
10±5
𝒇𝜽
0
0
0
0.005±0.005
0.005±0.005
0.010±0.005
𝒅𝜽 (𝒏𝒎)
---10±5
10±5
10±5
Fig. ‎3-3(a) Combined IQ and IPF of the QT-200 specimen, (b) a selected region in Fig. 3-3a in that the white and
red lines indicate the prior austenite grain boundaries and packet boundaries, respectively. The black lines are
high angle boundaries with misorientation higher than 15°, (c) distribution of martensite block sizes in the
same specimen.
The adapted fraction and particle size of 𝜀-carbides, cementite as well as mass fraction of
carbon in solid solution is displayed in Table ‎3-1. The particle size of 𝜀-carbide is estimated
from SEM micrographs and for the cementite particles is assumed the same as 𝜀-carbide.
38
Chapter 3
Fig. ‎3-4 (a) The engineering stress-strain curves and (b) the yield strength of the as-quenched as well as
tempered specimens.
Fig. ‎3-5 Contribution of a strengthening mechanisms on the total yield strength of the as-quenched as well as
tempered specimens. For each specimen, the solid solution strengthening from carbon atoms (𝜎𝐶 ), lattice
friction stress for pure Fe (𝜎0 ), the dislocation strengthening (𝜎𝜌 ), precipitation strengthening from 𝜀-carbide
𝜀
𝜃
(𝜎𝑝𝑐𝑝𝑡
), precipitation strengthening from cementite (𝜎𝑝𝑐𝑝𝑡
), the solid solution strengthening from substitutional
atoms (𝜎𝑠𝑡 ), grain boundary strengthening (𝜎𝑔𝑏 ) and dislocation strengthening (𝜎𝜌 ) are shown sequentially in
the columns.

Martensite morphology
The martensite laths are finer than the resolution of EBSD and cannot be detected by
EBSD. The martensitic blocks were successfully indicated from EBSD micrographs by the
definition of boundary misorientation higher than 15° as the block boundary [38]. Fig. ‎3-3a
shows the combined image quality (IQ) and inverse pole figure (IPF) maps resulting of EBSD
analysis of the specimen QT-200. An initial austenite grain is identified as an area in which
the {110} pole figure of austenite contains three reflections and is shown in Fig. ‎3-3b.
The black lines in Fig. ‎3-3b represent boundaries with misorientation higher than 15° and
martensite blocks can thus be identified in this image. The martensite packets define a group
of laths with similar habit planes and the martensite laths in a packet have the same
orientation relationship with the prior austenite, based on parallel closed-packed planes (a
CP group) [39].
39
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Fig. ‎3-6 The dislocation density determined based on the dislocation strengthening (Eq. ‎3-13), the MWH
method (Eq. ‎3-2) and the combined MWH-MWA method (Eq.‎3-12) are shown by cubic, triangle and
diamond symbols, respectively. The uncertainty of the dislocation densities based on dislocation
strengthening, the MWH, the combined modified MWH-MWA approach are 2 × 1015 m−2 , 2 × 1014 m−2
and 3 × 1015 m−2 , respectively.
Martensite packets within an initial austenite grain are indicated with red lines in
Fig. ‎3-3b. The martensite block size was considered as the average size of grains with a
misorientation larger than 15°. The martensite block size distribution, which is shown in
Fig. ‎3-3c, has an average size of 0.4 μm. Similar average block size was also obtained for the
specimen QT-0. This indicates that no significant block coarsening occurs during the
tempering process.
The engineering stress-strain curves are represented in Fig. ‎3-4a. The yield strengths of
the specimens were determined by applying 0.2% offset method and are represented in
Fig. ‎3-4b. The yield strength increases sharply after 5 s of tempering process and it gradually
decreases during longer tempering process.
3.4.2 Calculation of the dislocation density from the yield strength
The solid solution strengthening, grain boundary strengthening and precipitation
strengthening are calculated in this section. Then, subtracting their contributions from the
total yield strength by using Eq. ‎3-13, the dislocation strengthening is determined.

Peierls-Nabarro force.
It is the stress which is required to move a dislocation through a perfect lattice. A value of
41 MPa is adopted by Speich and Swann [40] in their calculation for tempered martensite in
a steel containing 0.4 wt.% of carbon. In the current research, the Peierls-Nabarro force for
the as-quenched and tempered specimens is therefore presumed 41 MPa.

Solid solution strengthening.
For the specimen QT-0 (as quenched), the effect of carbon in solid solution strengthening
is neglected and only solid solution strengthening from substitutional elements is
considered. For the tempered specimens, the solid solution strengthening form carbon
atoms is calculated by replacing the assumed mass fraction of carbon, 0.05 wt.%, and
tempering time in Eq. ‎3-15.
40
Chapter 3
Fig. ‎3-7 X-ray diffraction profile of the specimen QT-0. The horizontal axis is based on the diffraction vector
2sin𝜃
and is derived using 𝐾 =
. A diffraction peak which is fitted with a Voigt function is shown.
𝜆
Fig. ‎3-8 Variations of diffraction peak width (∆K/FWHM) plotted against isothermal holding time. Error of ∆K is
about 0.002 nm.
Solid solution strengthening from carbon in the specimens QT-5, QT-10, QT-50, QT-100
and QT-200 are determined as 198 MPa, 130 MPa, 17 MPa, 3 MPa and 1 MPa, respectively,
and presented in Fig. ‎3-5. Substituting the chemical composition of the steel in Eq. ‎3-16, the
solid solution strengthening from substitutional atoms is calculated, the results are shown in
Fig. ‎3-5. Note concentrations of the substitutional atoms are the same in all the specimens
and therefore 𝜎𝑠𝑠 is the same, 294 MPa.

Grain boundary strengthening.
Given the block size of 0.4 m from EBSD analysis, the contribution of grain boundary
strengthening is determined by using Eq. ‎3-17 and is shown in Fig. ‎3-5. Since no grain
coarsening occurred during isothermal holding, all the tempered as well as as-quenched
specimens have the same 𝜎𝑔𝑏 as 332 MPa.

Precipitation strengthening.
The contribution of the 𝜀-carbide strengthening is given by replacing the average particle
size (𝑑𝜀 ) and the fraction of 𝜀-carbide (𝑓𝜀 ) of the specimens in Eq. 17. The particles sizes and
fractions are listed in Table ‎3-1.
As it is shown in Fig. ‎3-5, the contribution of the 𝜀-carbide precipitation strengthening on
the yield strength of the QT-0 specimen and the QT-5 specimen is 267 MPa and 205 MPa,
respectively, and for the QT-50, QT-100 and QT-200 specimens is 132 MPa.
For the specimens QT-50, QT-100 and QT-200, the diameter of cementite precipitates
(𝑑𝜃 ) is considered as 10 nm and the volume fraction of cementite (𝑓𝜃 ) is considered as
0.004, 0.006 and 0.001, respectively.
41
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Table ‎3-2 Parameters to calculate q and 𝐶ℎ00 in BCC crystal [17] with elastic parameters of 𝐶11 , 𝐶12 and 𝐶44 as
230 GPa, 135 GPa and 116 GPa, respectively [36].
𝑪𝒉𝟎𝟎
q
Parameters
screw
edge
screw
edge
𝒂
8.659
7.2361
0.1740
1.6690
𝒃
0.3730
0.9285
1.9522
21.124
𝒄
0.0424
0.1359
0.0293
0.0
𝒅
-6.074
-5.7484
0.062
0.0757
Fig. ‎3-9 Plot of Eq. ‎3-5 after some iterations and q have been converged. The reciprocal of the intersect on the
2
H axis gives q.
The contribution of the cementite precipitation strengthening on the yield strength is
calculated by using Eq. ‎3-19. As shown in Fig. ‎3-5, the precipitation strengthening is 176, 205
and 287 MPa for the specimens QT-50, QT-100 and QT-200, respectively.

Dislocation strengthening.
Substituting the numerical values of 𝜎0 , 𝜎𝑠𝑠 , 𝜎𝑔𝑏 , 𝜎𝑝𝑐𝑝𝑡 into Eq. ‎3-13, the contribution of
the dislocations on the yield strength, 𝜎𝜌 , can be approximated. Results are shown in
Fig. ‎3-5. The dislocation densities of the specimens were obtained from Eq. ‎3-20 and given
𝑀𝑑 =3 [41], 𝛼𝑑 =0.25 [2], 𝐺=80 GPa [42] and 𝑏=0.248 nm for iron [12]. The calculated
dislocation densities are presented in Fig. ‎3-6, together with the values calculated from XRD,
which will be presented in section ‎3.3.3.
It should be mentioned that for the tempered specimens all the assumptions made in the
amount of carbon in solid solution, fraction of carbide as well as their particle sizes, cause an
average uncertainty of about 20%. In the case of the specimen QT-0, assuming even 0.05
wt.% of carbon in solid solution gives a dislocation density in the order of 6 × 1014 m−2
which is beyond the range of reported dislocation density of the martensitic steel with 0.3
wt.% carbon. Accordingly, it is correct to assume that the amount of carbon in solid solution
in the specimen QT-0 is zero. After 5 s of the tempering process, there is a sharp drop in the
dislocation density and consequently in its contribution to the yield strength.
42
Chapter 3
Table ‎3-3 The values of 𝑞, the effective outer cut-off radius of dislocation (𝑅𝑒 ), the dislocation distribution
parameter (𝑀) and crystallite size (D).
Specimen
QT-0
QT-5
QT-10
QT-50
QT-100
QT-200
𝒒 ± 𝟎. 𝟎𝟐
2.20
2.27
2.34
2.27
2.33
2.31
𝑹𝒆 ± 𝟐(𝐧𝐦)
16
30
35
34
45
39
𝑴 ± 𝟎. 𝟏
1.2
1.5
1.7
1.6
1.8
1.7
𝑫 ± 𝟎. 𝟓 (𝛍𝐦)
0.2
0.4
0.4
0.5
0.6
0.5
Fig. ‎3-10 The modified Williamson-Hall plot (Δ𝐾 vs.𝐾𝐶̅ 1/2 ) for the specimen QT-0 (as-quenched).
The increase of the yield strength during this time interval could be related to the
dissolution of ɛ-carbide that provides carbon in solid solution and consequently increases
the solid solution strengthening. According to the current calculations, between 5 s and 200
s of isothermal holding time, the dislocation density does not change significantly.
3.4.3 Calculations of dislocation density from XRD peak broadening
The X-ray diffractogram of the specimen QT-0 is shown in Fig. ‎3-7, evidencing BCC
martensite reflections. However, some traces of FCC austenite can be seen in the diffraction
pattern of the tempered specimens. The volume fraction of the detected austenite is less
than 0.02 and does not influence the current research.
As it is shown in Fig. ‎3-7, the diffraction peaks are fitted with Voigt functions. Fig. ‎3-8
shows variations of the peak width of all reflections during isothermal holding, after
removing the instrumental broadening using Eq. ‎3-22 and Eq. ‎3-23. There is a sharp drop in
the peak width after 5 s of the tempering process. Further isothermal holding leads to an
insignificant peak-width reduction. This agrees well with decreasing the dislocation densities
which were calculated based on the yield strength in Fig. ‎3-6. The calculation of the
dislocation contrast factor is the first step in the computational procedure of the dislocation
density based on the MWH and MWA methods. In order to determine 𝐶̅ , the parameter q
was derived, on the basis of Fig. ‎3-5, from a linear relation between (∆𝐾 − 𝛼)2 /𝐾 2 and 𝐻 2 .
The plot to give the 𝑞-value for the QT-0 specimen is shown in Fig. ‎3-9.
43
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Fig. ‎3-11 The modified Warren-Averbach plot of the specimen QT-0 (as-quenched).
The optimal value of 𝛼 is determined as 3 × 10−4 nm−1 and from the intercept at the
horizontal scale the value of 𝑞 was determined as 2.2. The experimental value of q for the
other specimens is listed in Table ‎3-3.
The fractions of the screw and edge dislocations are calculated by using Eq. ‎3-6 and
considering the theoretical value of q for pure screw and edge as 2.7 and 1.3, respectively.
The theoretical values of q are obtained by substituting required parameters from Table ‎3-2
in Eq. ‎3-7 and taking the elastic constants of the steel, i.e. 𝐶11 , 𝐶12 and 𝐶44 as 230 GPa, 135
GPa and 116 GPa, respectively [36].
From the numerically calculated fractions of screw dislocations, it follows that more
dislocations (0.65-0.75) have a screw character than an edge character, which agrees well
with TEM observations in BCC iron [43] and ferritic or martensitic steels [44]. In this research
the fraction of screw dislocations is considered 0.7 which is an intermediate value between
0.65 and 0.75. The values of the parameter Ch00 for pure screw and edge dislocations,
obtained via solving Fig. ‎3-5 and using the parameters given in Table ‎3-2, are 0.16 and 0.26,
respectively. Considering that fractions of screw and edge dislocations in the microstructures
are 0.70 and 0.30, respectively, then the value of ̅̅̅̅̅̅
𝐶ℎ00 leads to 0.19. Next, 𝐶̅ for each
diffraction peak was calculated by applying Eq. ‎3-3.
The dislocation density based on the MWH method was determined by fitting Eq. ‎3-2
with the experimental values ∆𝐾 versus 𝐾𝐶̅ 2 and given a fixed value to M. There are some
controversies in the literature about the value of M. While in [6, 7], the MWH method is
applied assuming M as 0.1, it has been shown in [5] that the MWH method is applicable for
values of M higher than 1. The controversy is due to the fact the method is valid as long as
the actual dislocation distribution of the deformed crystal does not deviate too much from
the assumed model and as long as the distribution parameter of dislocations is not too small
(i.e. 𝑀 > 1) [5]. In this work, 𝑀 is considered 1.4 by using the value given in [45]. In that
research, 𝑀 was calculated from 𝑀 = 𝑅𝑒 √𝜌 in which 𝑅𝑒 and 𝜌 are obtained from the MWA
method. To exemplify the applied method, the MWH plot and fitted curve for the specimen
QT-0 is shown in Fig. ‎3-10.
44
Chapter 3
𝑋(𝐿)
Fig. ‎3-12 Plot of 2 versus L for the specimen QT-0. The open circle shows the experimental data, the solid line
𝐿
belongs to the fitted line for 𝑓(𝜂) in which 𝜂 < 1, the dotted line is fitted for 𝑓(𝜂) in which 𝜂 > 1. Moreover,
𝐿
the dash line represents the fitted curve by using Eq. ‎3-11 and assuming the strain function, 𝑓(𝜂), as Ln( ).
𝑅𝑒
The same procedure was applied for other specimens and the calculated values of the
dislocation density are given in Fig. ‎3-6. The MWH approach, under the assumption of an
appropriate value for M, gives an estimation of the dislocation density that is consistent with
the dislocation strengthening approach. This shows that the MWH method can be used as an
accessible method for dislocation density determination. Moreover, the approximated
values of 𝐷 are given in Table ‎3-3.
Fig. ‎3-11 shows the Modified Warren-Averbach plot of the specimen QT-0 that was
obtained by plotting ln𝐴(𝐿) versus 𝐾 2 𝐶̅ at different Fourier lengths, 𝐿. Noted, the real part
of the Fourier coefficients is obtained from Eq. ‎3-25. The MWA plot gives 𝑋(𝐿) in Eq. ‎3-10 as
the slope of the line. Fig. ‎3-12 shows
𝑋(𝐿)
𝐿2
versus 𝐿 for the specimen QT-0 specimen in which
𝑅
Eq. ‎3-11 is fitted to the experimental data by assuming 𝑓(𝜂) as Ln( 𝐿𝑒 ). The logarithmic
function is only fitted at very narrow range of L and therefore it does not give the real value
of 𝑅𝑒 and 𝜌. To overcome this difficulty, both MWH-MWA methods were combined in the
present work. This was done by fitting the experimental data points in Fig. ‎3-12 with Eq. ‎3-12
and considering the dislocation density as the fitting parameter. In this approach, 𝑓(𝜂∗ ) is
given by Eq. ‎3-9. The parameter of M was determined by replacing 𝛾 (from the MWH
method) in 𝛾 = 𝑀√𝜌. Knowing 𝑀 and 𝜌, the parameter of 𝑅𝑒 was given by 𝑀 = 𝑅𝑒 √𝜌. The
estimated values of 𝜌, 𝑅𝑒 and M are given in Table ‎3-3. The same procedure was applied for
the other specimens and the values of 𝜌, 𝑅𝑒 and M are given in Table ‎3-3. The uncertainty
range corresponds to the residue of the fitting of the diffraction peaks. The estimated value
of M from this approach is in the range of assumed value (1.4). According to [3], 𝑅𝑒 is related
to the scale of strain fields of dislocations. Accordingly, the increase of 𝑅𝑒 during the
tempering shows that the distribution of the dislocations becomes random and their
correlation is decreased. Fig. ‎3-6 presents the dislocation density of the specimens obtained
from MWH as well as the combined MWH-MWA methods.
45
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
Fig. ‎3-13 Dislocation density of the as-quenched steel evaluated based on the dislocation strengthening
(triangle), the MWH method (square), the combined MWH-MWA method (diamond) and measurement based
on TEM observations (circle and solid line). The TEM results were taken from [1] for Fe-C steels.
The calculated dislocation density from the combined MWA-MWH approach agrees well
with the values obtained from the dislocation strengthening approach. This indicates that
the developed approach can be considered as a quantitative method for the dislocation
density characterization.
Morito et al. [1] investigated the variation of the dislocation density against the carbon
content in as quenched Fe-C steels by transmission electron microscopy (TEM). To make a
comparison between dislocation density of the specimen QT-0 (as-quenched) and the value
obtained in [1], the difference in the martensite start temperature (Ms) will be considered in
the present work.
The studied steel, in addition to carbon contains other austenite stabilizer elements i.e.
Mn and Si in its chemical composition. Adding austenite stabilization alloying elements to
the chemical composition of steels decrease Ms. Steels with lower Ms, are harder and
therefore the shear stress associated with the formation of martensite and as a consequence
the plastic deformation, i.e. the dislocation density, is higher. Fig. ‎3-13 illustrates the
variations of the dislocation density of Fe-C steels obtained by TEM in [1] as a function of Ms,
in K, that was estimated by using [46]:
𝑀𝑠 = 838 − 600[1 − 𝑒𝑥𝑝(−0.96𝑋𝐶 )]
‎3-28
where 𝑋𝐶 is the concentration of C in wt.%. Moreover, dislocation density of the specimen
QT-0 derived on the basis of dislocation strengthening, MWH and combined MWH-MWA
methods are added to Fig. ‎3-13. For the studied steel, Ms is estimated from dilatometry data
as 560 K. According to Eq. ‎3-28, the Fe-0.65C steel has Ms of 560 K which is similar to the
studied steel. Dislocation density of the Fe-0.65C steel is estimated by TEM analysis about
3.2 × 1015 𝑚−2 and it is added to Fig. ‎3-13. It can be seen in that the dislocation density of
the Fe-0.65C steel is in the range of the values obtained from dislocation strengthening
(4.9 × 1015 m−2 ), MWH (5.5 × 1015 m−2 ) and MWA (5.9 × 1015 m−2 ) methods. In the
current research, the strain broadening of the X-ray diffraction profile is entirely related to
the dislocations, thereby other contributions such as presence of carbon in solid solution are
neglected. This could explain the slight overestimate of the dislocation density by the line
broadening analysis as compared to the other methods.
46
Chapter 3
3.5
Conclusions
In this chapter the dislocation density of a lath martensitic steel in the as-quenched as
well as tempered conditions was evaluated by applying different X-ray diffraction profile
analysis methods. Moreover, the dislocation density of the specimens were determined
based on the dislocation strengthening. The following conclusions were drawn:

At the initial stage of the tempering process, there is a drop in the dislocation
density while the yield strength increases. The increase of the yield strength could
be related to the increase of interstitial carbon solid solution strengthening that is
caused by decomposition of 𝜀- carbide. The 𝜀- carbide was formed in martensite
during the quenching.

The modified Williamson-Hall equation (MWH), under the assumption of a fixed
value for the dislocation distribution parameter (𝑀 = 1.4), was applied to
calculate the dislocation density. The calculated dislocation densities show
uncertainty of about 40% regarding the values obtained from the dislocation
strengthening. This approach can be considered as an accessible approach to
estimate the dislocation density of lath martensitic steel.

An improved method for calculating the dislocation density is developed by
combining the MWH and MWA methods. The proposed method is independent
from the range of the Fourier length (L). This method leads to a dislocation density
that is in good agreement with the dislocation density determined based on the
dislocation strengthening. It was found that the proposed method can be used as
a quantitative method for dislocation density calculations.
47
An Improved X-ray Diffraction Analysis Method to Characterize Dislocation Density
3.6
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50
CHAPTER 4
4 Microstructural Characterization of a
0.3C-1.6Si-3.5Mn (wt.%) Quenching and
Partitioning Steel*
Abstract
Theoretical understanding of the "quenching and partitioning" (Q&P) process allowed
developing microstructures consisting of carbon-depleted martensite and retained austenite
that deliver superior mechanical properties. Most of the models describing the Q&P process
are limited to systems in which carbide precipitation in martensite and decomposition of
austenite to bainite are totally suppressed. However, these reactions are often unavoidable,
even in low-carbon steels containing a relatively high concentration of Si and Mn. This work
investigates interactions between carbon partitioning, carbide precipitation and carbide-free
bainite formation during the Q&P process of a 0.3C-1.6Si-3.5Mn (wt.%) steel with nonhomogenous distribution of the alloying elements. It was found that prior to the partitioning
step ɛ-carbide forms in martensite. The decomposition of this carbide is required for a full
completion of the carbon partitioning from martensite to austenite. Slow kinetics of
decomposition of ɛ-carbide retards the carbon partitioning process. Results show that a
fraction of austenite becomes stable by carbon partitioning and does not decompose to
bainite. In the specimens quenched to lower temperature, a higher fraction of austenite
becomes stable and consequently a lower fraction of bainite is formed.
Keywords: Quenching and Partitioning, Low carbon steels, Carbides, Bainite, Microstructural
modelling
*
This chapter is based on a scientific paper:
F. HajyAkbary, J. Sietsma, G. Miyamoto, T. Furuhara, M. J. Santofimia, Interaction of Carbon Partitioning,
Carbide Precipitation and Bainite Formation during the Quenching &Partitioning Process in a low C steel,
submitted to Acta Materialia.
51
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
4.1
Introduction
The "quenching and partitioning" (Q&P) process is known as a promising method for
developing steels with good combinations of strength and ductility [1]. The Q&P process
involves rapid quenching of an austenitic microstructure to a temperature lower than the
martensite-start temperature (Ms) to form a controlled fraction of initial martensite (M1).
Then, the treatment is followed by isothermal holding either at or above the initial
quenching temperature to stabilize austenite via carbon partitioning from supersaturated
martensite to austenite. The Q&P process is ended by quenching the microstructure to room
temperature during which some secondary martensite (M2) may form if the carbon
enrichment is not sufficient to stabilize all austenite [2]. Secondary martensite contains a
high concentration of carbon and is detrimental for ductility [3]. Formation of M2 can be
controlled by knowledge of the carbon partitioning process as well as its interactions with
other possible reactions such as martensite-austenite interface migration, carbide
precipitation in martensite and decomposition of austenite to bainite.
The thermodynamics of the carbon partitioning process can be well described on the
basis of "constrained carbon equilibrium" (CCE) [4, 5]. In this approach, the partitioning
process across an immobile austenite-martensite interface is ended when martensite (i.e.
ferrite) is in metastable equilibrium with austenite. Santofimia et al. [6] adapted the model
to simulate the interaction between the carbon partitioning and the possible migration of
martensite-austenite interfaces. These approaches are limited to well-controlled systems in
which carbide precipitation in martensite and decomposition of austenite to bainite are
totally suppressed. However, these reactions are often unavoidable, even in low-carbon
steels containing a relatively high concentration of Mn and Si [7, 8, 9].
Precipitation of ɛ-carbide prior to the partitioning step or of cementite during the
isothermal holding affects the carbon partitioning process. Precipitation of cementite
reduces the amount of carbon in solid solution in martensite and therefore decreases the
degree of carbon enrichment that can be reached in austenite [10, 11]. Generally, a high
concentration of Si or Al is added to the steel composition to control cementite formation
[12], however, these elements increase the stability of the ɛ-carbides [13]. Consequently,
suppressing precipitation of ɛ-carbide is really challenging and practical designing of Q&P
treatments requires knowledge of the interaction between ɛ-carbide precipitation and
carbon partitioning process.
The carbon partitioning process may also overlap with the decomposition of austenite to
bainite. Formation of carbide-free bainite associates with carbon enrichment of austenite
and could stabilize austenite. Therefore, bainite formation has an important influence on the
final microstructure. Developing a model that indicates the interaction between bainite
formation and carbon partitioning process assists in better controlling the microstructure.
In this paper, the microstructural evolution during the Q&P process of a 0.3C-1.6Si-3.5Mn
(wt.%) steel with non-homogenous chemical composition is analysed. The influence of the
carbide precipitation as well as the formation of carbide-free bainite on the microstructure is
discussed based on the experimental observations and the microstructural modelling.
52
Chapter 4
4.2
Experimental procedures
Cylindrical specimens with length of 10 mm and diameter of 3.5 mm were machined
parallel to the hot-rolling direction of 0.3C-1.6Si-3.5Mn (wt.%) steel sheets. A scheme of the
applied heat treatments is shown in Fig. 1a. The specimens were austenitized at 900 °C for
180 s, quenched to 180 °C, 200 °C, 220 °C, 240 °C and 260 °C, isothermally treated at 400 °C
for 5 s, 10 s, 50 s, 100 s and 200 s and finally quenched to room temperature in a Bähr DIL
805 A/D dilatometer. In this paper, the code QTxxx-y identifies the specimen that was
quenched to xxx °C and isothermally treated at 400 °C for y seconds. In addition to the Q&P
specimens, one "as-quenched" specimen was created by austenitization at 900 °C for 180 s
and then directly quenched to room temperature.
After conventional metallographic preparation, specimens were etched with 2% Nital for
subsequent optical microscopy and scanning electron microscopy (SEM) observations using a
JEOL JSM-6500F field emission gun scanning electron microscope (FEG-SEM) operating at 15
kV. The specimens were metallographically prepared for electron backscatter diffraction
(EBSD) examination with a final polishing step of 0.05 μm using an OPS suspension for 1
hour. The analyses were done by orientation imaging microscopy (OIM) on a FEI Nova 600
Nanolab dual-beam (focused ion beam) FEG-SEM, under the following conditions:
acceleration voltage 20 kV; working distance 25 mm; tilt angle 70°; step size 50 nm. The
orientation data were post-processed with the TSL system. Furthermore, selected specimens
were observed with a transmission electron microscope (TEM; Philips CM300) operated at
300 kV. Thin-foils of TEM were prepared by twin-jet electropolishing at 25-30 V in 10% HClO4
in ethanol solution at room temperature. Distributions of Mn and Si in selected regions were
analysed using electron probe micro analyser (EPMA) technique. EPMA measurements were
performed with a JEOL JXA 8900R microprobe using a 10 keV electron beam with beam
current of 50 nA employing wavelength-dispersive spectrometry (WDS).
Volume fractions of RA (𝑓 RA ) as well as carbon concentrations of RA (𝑋CRA ) were
determined by means of X-ray diffraction (XRD) analysis using a Bruker type D8-Advance
diffractometer, in a 2θ range from 30° to 135°, with Co Kα radiation. The calculation of 𝑓 RA
and 𝑋CRA was performed in accordance with the method given in [4]. Volume fractions of
initial martensite (𝑓 M1 ), formed during the initial quench, bainite (𝑓 B ), formed during the
isothermal holding, and secondary martensite (𝑓 M2 ), formed during the final quench, were
evaluated by applying the lever rule on the dilatometer data (Fig. 1b).
4.3
Results
4.3.1 Analysis of compositional gradients in the steel
Fig. ‎4-2a shows an optical micrograph of the specimen QT260-5. The microstructure
consists of internally etched features, M1, and blocky white features, M2, which are not
homogenously distributed. In this specimen, as in the other Q&P specimens, microstructural
bands are parallel to the rolling direction. This shows non-homogeneous distribution of the
alloying elements in the transverse direction. Concentrations of Mn and Si along the
transverse direction, the black arrow in Fig. ‎4-2a, were determined by EPMA and the results
are illustrated in Fig. ‎4-2 b. Concentrations of Mn and Si fluctuate between 2.9-4.2 wt.% and
1.1-1.6 wt.%, respectively.
53
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎4-1 (a) Scheme of the heat treatments and (b) dilatometer curve of the specimen QT220-200.
Non-homogeneous distribution of the Mn and Si is caused by the rejection of these
elements to the inter-dendritic spaces during the solidification process. According to Fig. ‎4-2,
the fraction of M1 is higher in the low-solute regions and the fraction of M2 is higher in the
high-solute regions. Regarding the fact that Mn lowers the chemical potential of carbon in
austenite, it is expected that high-Mn regions attract carbon. The influence of Mn
segregation on carbon concentration at the austenitization temperature (900 °C) is
calculated by ThermoCalc, using TCFE7 data base. According to the calculations, increasing
Mn concentration from 2.9 wt.% to 4.2 wt.% increases the carbon concentration from 0.29
wt.% to 0.31 wt.%. This was done by determining the equilibrium carbon concentration at
the austenitization temperature (900 °C) while varying the Mn concentration. In these
calculations, Mn concentration varies in the range of 2.9-4.2 wt.% with a step of 0.25 wt.%
and the concentration of Si is considered fixed (1.6 wt.%).
4.3.2 Microstructural observations
Fig. ‎4-3a-3d show SEM micrographs of the as-quenched, QT180-5, QT220-5 and QT260-5
microstructures, respectively. The as-quenched specimen consists of martensite and
carbides. The Q&P specimens are composed of internally etched features, which are related
to M1 and blocky features which are related to M2 and thin films of RA. Precipitation of
carbide inside M1 grains and thin films of RA are also evident.
Fig. ‎4-4a and 4b show bright field TEM images of the specimens QT180-5 and QT180-200,
respectively. The density of carbides in the specimen QT180-200 is lower than in the
specimen QT180-5. The carbide type was identified by selected area diffraction (SAD)
analysis; here only the ε-carbide and cementite were considered. Fig. ‎4-4c illustrates SAD
pattern and Fig. ‎4-4d the ideal SAD pattern corresponding to the bright field image of the
specimen QT180-200. Table ‎4-1 compares the calculated and the experimentally obtained
interplanar spacing between vector 1 and 2 in Fig. ‎4-4c. The measured interplanar spacing is
close to the values reported for ε-carbide, which confirms the presence of ε-carbide in the
specimens.
Fig. ‎4-5 shows a combined grain average Image Quality (IQ) and phase map of the
specimen QT220-5 obtained by EBSD. In this map, RA grains are shown in green and
martensite grains, including M1 and M2, are indicated in red. Due to the higher carbon
concentration and dislocation density of M2 with respect for M1, the IQ for M2 grains is lower
than of M1 grains [15]. Some M2 grains, dark red grains, are indicated in Fig. ‎4-5.
54
Chapter 4
Fig. ‎4-2 (a) Optical micrograph of the specimen QT260-5 and (b) distribution of Mn and Si along the black
arrow.
4.3.3 Modelling of carbon partitioning process
Carbon partitioning from martensite to austenite, at the isothermal holding temperature
of 400 °C, is simulated based on the model given in [6] for a Fe-C system with immobile
martensite-austenite interfaces. The model assumes that carbide precipitation and bainite
formation do not occur before or during the isothermal holding. The lath width of
martensite is assumed independent of the quenching temperature and given as 0.2 μm [16].
The austenite grain size is determined using the "constant-ferrite width" approach [17] as
0.05 μm and 0.8 μm for the specimens QT180 and QT260, respectively, because of the
respective austenite fractions at the quenching temperature.
Fig. ‎4-6a and 6b illustrate the evolution of the carbon profile in an austenite grain of the
specimens QT180 and QT260, respectively. Based on the simulations, isothermal holding of 1
s is sufficient for complete carbon diffusion from martensite to austenite for all austenite
fractions after the quenching. However, 5 s of isothermal holding is sufficient to homogenise
carbon inside an austenite grain of the specimen QT180, isothermal holding longer than 200
s is required for the specimen QT260. This results from the low mobility of carbon in
austenite and regarding the fact that the considered austenite grain size for the specimen
QT260 is about 16 times of the specimen QT180.
The critical carbon concentration by which austenite becomes stable at room
temperature can be determined on the basis of the relation between chemical composition
and Ms (°C)[18]:
𝑀𝑠 = 565 − 600[1 − exp(−0.96𝑋𝐶 )] − 31𝑋𝑀𝑛 − 13𝑋𝑠𝑖 − 10𝑋𝐶𝑟 − 18𝑋𝑁𝑖 −
12𝑋𝑀𝑜 ,
‎4-1
where 𝑋𝑖 is the concentration of the element 𝑖 in wt.%. Considering 𝑀𝑠 as 25 °C, 𝑋𝑀𝑛 as
3.5 wt.% and 𝑋𝑆𝑖 as 1.6 wt.%, the critical carbon concentration of austenite is given as 1.2
wt.%. Accordingly, austenite with carbon concentration lower than 1.2 wt.% will transform
to M2 during the final quenching. Fig. ‎4-6a shows that after 5 s of isothermal holding,
calculations show that austenite in the specimen QT180 contains 1.5 wt.% carbon and
therefore all austenite retains at room temperature. On the contrary, Fig. ‎4-6b shows that a
steep carbon gradient from 0.95 wt.% to 0.3 wt.% is observed in calculations correspond to
austenite in the specimen QT260-5.
55
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎4-3 SEM micrograph of the specimens (a) as-quenched, (b) QT180-5, (c) QT220-5 and (d) QT260-5. M1 is
initial martensite, RA is retained austenite and M2 is secondary martensite.
After 200 s of isothermal holding of the specimen QT260, the carbon distribution
becomes almost homogeneous within the austenite grain and it decreases to 0.5 wt.%. The
carbon concentration of austenite in both the specimens QT260-5 and QT260-200 is below
the critical concentration and therefore part of austenite transforms to M 2 during the final
quenching.
4.3.4 Volume fractions and carbon content of retained austenite
Volume fractions of RA (𝑓 𝑅𝐴 ) in the Q&P specimens are determined on the basis of XRD
analysis and the carbon partitioning simulations. The results are shown in Fig. ‎4-7a. A
comparison between the calculated and the measured 𝑓 𝑅𝐴 leads to the following
observations;
(a) For each partitioning time, the measurements show that the specimens QT200 and
QT220 have the highest fractions of RA. Considering the experimental uncertainty, this
agrees well with the simulations of carbon partitioning process ( section 4.3.3) which give
the highest fraction of RA in the specimens QT200.
(b) According to the simulations by increasing the isothermal holding time from 5 s to 200
s, 𝑓 𝑅𝐴 of the specimens QT180 and QT200 does not change. However, simulation predicts
that increasing the isothermal holding time from 5 s to 200 s leads to the reduction of 𝑓 𝑅𝐴 in
the specimens QT220, QT240 and QT260 decreases.
56
Chapter 4
Table ‎4-1 Comparison between the calculated and experimentally obtained interplanar spacing of line 1
and 2 in Fig. ‎4-4c. The 𝑑-spacing was calculated using the parameters 𝑎=2.752 Å, 𝑐=4.353‎Å‎for‎ε‎–
carbide [19] and 𝑎=4.525 Å, 𝑏= 5.900 Å, 𝑐=6.744‎Å‎for‎θ‎[20].
Measured interplanar spacing (Å)
Calculated interplanar spacing (Å)
ε
θ
1
4.36
𝑑0001 =4.35
𝑑011 =4.06
2
2.40
𝑑11̅00 =2.38
𝑑200 =2.26
Fig. ‎4-4 Bright field TEM micrograph of the specimens (a) QT180-5 and (b) QT180-200, (c) SAD pattern of region
shown in Fig. ‎4-4b and the corresponding key diagram. Filled circles belong to ferrite reflections and open
circles show carbide reflections and beam~// [100]α~//[112̅0]ε.
On the contrary, the measured 𝑓 𝑅𝐴 increases by increasing the isothermal holding time
from 5 s to 200 s, being the increase of 𝑓 𝑅𝐴 more pronounced in the specimens QT220,
QT240 and QT260.
(c) For all the quenching temperatures, the simulated 𝑓 𝑅𝐴 of the specimens isothermally
holded for 5 s is higher than the measured ones. By increasing the isothermal holding time
to 200 s, the simulated 𝑓 𝑅𝐴 of the specimens QT180 and QT200 is still higher than the
measured ones. However, the simulated 𝑓 𝑅𝐴 of the specimens QT220, QT240 and QT260 is
lower than the measured 𝑓 𝑅𝐴 .
57
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎4-5 Combined average grain Image Quality (IQ) and phase map of the specimen QT220-5. The black
boundaries are high angle grain boundaries (higher than 15°) and martensite and RA are coded by red and
green, respectively.
Carbon concentrations of RA (𝑋𝐶𝑅𝐴 ) are presented in Fig. ‎4-7b. The values of 𝑋𝐶𝑅𝐴 varies in
the range of 0.8 wt.%-1.0 wt.% which is lower than the value (1.2 wt.%) assumed in the
carbon partitioning simulation. This is due to the fact that the simulations do not consider
the influence of austenite grain size on Ms. By applying the method developed in [21] and
given the austenite grain size as it was determined in section ‎4.3.3, the critical carbon
concentration of austenite is estimated 0.9 wt.%. Furthermore, the lower carbon
concertation of RA can be justified by considering in-homogenous distribution of Mn and in
view of the fact that austenite grains with high Mn concentration requires lower carbon
concentration to become stable. No clear trend can be detected in the behaviour of 𝑋𝐶𝑅𝐴 of
the Q&P specimens as a function of the isothermal holding time and the quenching
temperature. Fig. ‎4-7c shows the effect of the isothermal holding time on the total carbon
content of RA (𝐶 𝑅𝐴 ). The total carbon content for a given phase (𝐶𝑗 ) is defined as:
𝑗
𝐶𝑗 = 𝑓 𝑗 𝑋𝐶 ,
‎4-2
𝑗
where 𝑓 𝑗 and 𝑋𝐶 are volume fraction and carbon concentration for phase j (j= RA, M1 and
M2), respectively. For the specimens QT180, QT200 and QT220, 𝐶 𝑅𝐴 increases during 50 s
and then remains constant. The increase of 𝐶 𝑅𝐴 of the specimens QT240 and QT260
continues during 100 s and 200 s of isothermal holding, respectively.
4.3.5 Dilatometry analysis of the Q&P heat treatments
The Q&P process is divided into three stages that are shown by numbers 1 to 3 in Fig. 1a.
Stage 1 (initial quench). The fractions of initial martensite (𝑓 𝑀1 ) that are formed during
the initial quenching of the specimens QT180, QT200, QT220, QT240 and QT260 are 0.79,
0.74, 0.65, 0.52 and 0.49, respectively. The fractions of M 1 are determined by applying lever
rule and the accuracy of the fraction is 0.02.
Stage 2 (isothermal holding). Fig. ‎4-8a shows the influence of the isothermal holding on
the relative length of the Q&P specimens. The increase of the relative length of the
specimens during the isothermal holding can be related to the carbon partitioning from
martensite to austenite [22] and to decomposition of austenite to bainite.
58
Chapter 4
Fig. ‎4-6 Evaluation of carbon profile in an austenite grain during isothermal holding of the specimens (a) QT180
and (b) QT26, on the basis of calculations.
The former reaction leads to a length increase that is much smaller than measured ones
[22] and therefore the total length increase is mainly related to the latter reaction. The
fraction of bainite (𝑓 𝐵 ) is determined by applying the lever rule and the results are shown in
Fig. ‎4-8b. For the specimens QT180 and QT200, 𝑓 𝐵 is less than 0.01, which is below the
detecting limit. For the specimens QT220, QT240 and QT260, 𝑓 𝐵 increases till 200 s.
Stage 3 (final quench). The expansion of the Q&P specimens during the final quench is
related to the transformation of unstable austenite to M 2. Fig. ‎4-9 shows volume fraction of
M2 (𝑓 𝑀2 ) versus isothermal holding time. For the specimens QT180 and QT200, isothermal
holding longer than 5 s does not affect 𝑓 𝑀2 . However, 𝑓 𝑀2 of the specimens QT220, QT240
and QT260 decreases by increasing the isothermal holding time. For a given isothermal
holding time, 𝑓 𝑀2 of the specimens QT180 and QT200 are similar, while 𝑓 𝑀2 of the
specimens QT220, QT240 and QT260 increase as a result of increasing the quenching
temperature.
4.3.6 Carbon content of secondary martensite
Carbon concentration of M2 (𝑋𝐶𝑀2 ) can be determined by fitting Koistinen and Marburger
(KM) equation [23], as:
𝑓 𝑀2 = 1 − 𝑒𝑥𝑝( −𝛼𝑚 (𝑇𝐾𝑀 − 𝑇)),
‎4-3
to the experimental plot of volume fraction of M2 (𝑓𝑀2 ) vs. quenching temperature (𝑇). In
this approach, αm (the rate parameter) and TKM (the theoretical martensite start
temperature) are fitting parameters. Finally, 𝑋𝐶𝑀2 is given by the empirical relations between
the chemical composition and αm as well as TKM [24]. This approach is applicable if carbon is
homogenously distributed in the austenite that is not sufficiently stabilised. However, it
might be that specimens are quenched before complete carbon homogenisation inside
austenite grains has taken place. Moreover, in steels with inhomogeneous chemical
composition (such as current study) the carbon concentration of unstable austenite
decreases by moving from low-Mn regions to high-Mn regions. Low-Mn regions have higher
fraction of M1 than high-Mn regions, therefore unstable austenite in the low-Mn contains a
higher carbon concentration. More details are given in section ‎4.4.1.
59
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎4-7 (a) Calculated (Calc.) and experimentally (Exp.) measured volume fractions of RA as a function of the
quenching temperature for the specimens isothermally holed at 400 °C for different times. Influence of
isothermal holding on (b) carbon concentration in RA (𝑋𝐶𝑅𝐴 ) and (c) total carbon content of RA (𝐶 𝑅𝐴 = 𝑓 𝑅𝐴 𝑋𝐶𝑅𝐴 )
in the Q&P specimens with different quenching temperatures.
Note that the simulations procedure is simplified by assuming concentrations of
substitutional elements as the nominal composition. In order to analyse the unstable
austenite with inhomogeneous carbon distribution, the unstable austenite is divided into
sub-fractions with homogenous composition. This means that the sum of the volume
fractions of unstable austenite with constant carbon concentration (𝑓 𝑖 ) is equal to the total
fraction of unstable austenite (𝑓 𝑈𝑛𝑆𝐴 ) as following:
𝑓 𝑈𝑛𝑆𝐴 = ∑𝑓 𝑖 ,
‎4-4
Accordingly, a combination of KM curves can simulate formation of M 2:
𝑖
𝑖
𝑓 𝑀2 = ∑𝑖 𝑓 𝑖 {1 − 𝑒𝑥𝑝 (−𝛼𝑚
(𝑇𝐾𝑀
− 𝑇))},
‎4-5
𝑖
𝑖
𝑖
𝑖
with 𝛼𝑚
and 𝑇𝐾𝑀
are the KM parameters for segment 𝑖. The parameters 𝛼𝑚
(K-1) and 𝑇𝐾𝑀
(°C) depend on the chemical composition according to [24]:
𝑖
𝛼𝑚
= 0.0224 − 0.0107𝑋𝐶𝑖 − 0.0007𝑋𝑀𝑛 − 0.00005𝑋𝑁𝑖 − 0.00012𝑋𝐶𝑟 −
0.0001𝑋𝑀𝑜 ,
60
‎4-6
Chapter 4
Fig. ‎4-8 The influence of the isothermal held at 400 °C on the (a) relative length change of the Q&P specimens
and (b) volume fractions of bainite in the specimens QT220, QT240 and QT260. The fractions of bainite are
almost zero in the specimens QT180 and QT200.
𝑖
𝑇𝐾𝑀
= 462 − 273 𝑋𝐶𝑖 − 26𝑋𝑀𝑛 − 16𝑋𝑁𝑖 − 13𝑋𝐶𝑟 − 30𝑋𝑀𝑜 ,
‎4-7
where 𝑋𝐶𝑖 (wt.%) is carbon concentration in the segment 𝑖 of unstable austenite and 𝑋𝑗
(wt.%) is concentration of element j (=Mn, Ni, etc.). Assuming that the carbon distribution
follows a Gaussian function, the carbon distribution in M 2 can be determined by inserting
Eq. ‎4-6 and Eq. ‎4-7 in Eq. ‎4-5 and considering the experimental plot of 𝑓 𝑀2 versus T. The
Gaussian distribution of carbon is determined by three optimisation variables: the peak
height (𝑓 𝑖 ), position of the centre of the peak (μ) and the standard deviation (σ). The
optimisation problem is solved using "MATLAB Optimisation" toolbox. Austenite with carbon
concentration higher than 0.9 wt.% is stable, thus the optimisation problem has one
constraint; the maximum carbon concentration of unstable austenite is 0.9 wt.%, i. e. 0< μ
<0.9 wt.%, see section ‎4.3.3. Another constraint is that the total fraction of unstable
austenite (𝑓 𝑈𝑛𝑆𝐴 ) in Eq. ‎4-4 is given by 𝑓 𝑈𝑛𝑆𝐴 = 𝑓 𝑀2 = 1 − 𝑓 𝑀1 − 𝑓 𝐵 − 𝑓 𝑅𝐴 . Here, 𝑓 𝑅𝐴 is
considered as the volume fraction of stable austenite.
Fig. ‎4-9b represents the carbon distribution in M2 of the specimens QT260 which was
obtained by applying this method. During the isothermal holding, the Gaussian distribution
of carbon moves toward higher concentrations, except when increasing the partitioning time
from 5 s to 10 s. Similar tendency to move the Gaussian distribution of carbon toward higher
concentrations is observed in the specimens QT220 and QT240. The carbon distributions in
M2 of the specimens QT180 and QT200 do not change significantly during the isothermal
holding.
The total carbon content of M2 (CM2) can be determined by applying Eq. ‎4-2 as 𝐶 𝑀2 =
∑𝑖 𝑓 𝑖 𝑋𝐶𝑖 . Fig. ‎4-9b shows the influence of the isothermal holding time on 𝐶 𝑀2 . While 𝐶 𝑀2 of
the specimens QT180 and QT200 is almost independent of the holding time, 𝐶 𝑀2 of the
specimens QT220, QT240 and QT260 decreases after 50 s of isothermal holding. At constant
isothermal holding time, 𝐶 𝑀2 increases by increasing the quenching temperature, however,
increasing the quenching temperature from 180 °C to 200 °C does not influence 𝐶 𝑀2 .
61
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Volume fraction of M2
0.1
(b)
QT260-5
QT260-50
QT260-100
0.08
QT260-10
0.06
0.04
QT260-200
0.02
0
0.25
0.3
0.35
0.4
0.45
Distribution of X
0.5
M2
C
0.55
0.6
0.65
(wt.%)
Fig. ‎4-9 The effect of the isothermal holding time on (a) volume fractions of M 2, (b) distribution of C in M2 of
the specimens QT260, (c) total carbon content of M2.
4.3.7 Carbon content of initial martensite
Carbon in M1 may precipitate as ɛ-carbide, segregate to defects or present in solid
solution. The total carbon content of M1 (𝐶𝑀1 ) can be determined by subtracting the carbon
content of RA (𝐶 𝑅𝐴 ) and the carbon content of M2 (𝐶 𝑀2 ) from the carbon content of the
steel (𝐶) as:
𝐶 𝑀1 = 𝐶 − 𝐶 𝑅𝐴 − 𝐶 𝑀2 .
‎4-8
Due to the low carbon solubility in ferrite, the carbon content of bainitic ferrite is
assumed zero. Considering the average carbon content of the steel (0.3 wt.%), 𝐶 𝑀1 of the
Q&P specimens were calculated with Eq. ‎4-8 and presented as a function of the isothermal
holding time in Fig. ‎4-10. The time to reach a constant 𝐶𝑀1 is 5 s for the specimens QT220,
QT240 and QT260. This time increases to 50 s for the specimens QT200 and QT180. For a
constant partitioning time, 𝐶 𝑀1 decreases by increasing the quenching temperature.
62
Chapter 4
Fig. ‎4-10 The effect of isothermal holding on the total carbon content of M1 (𝐶𝑀1 = 0.3 wt. % − 𝐶𝑀2 − 𝐶 𝑅𝐴 ).
4.4
Discussion
4.4.1 Effect of the non-homogeneity of the chemical composition
According to Fig. ‎4-2a and 2b, higher fractions of M1 are formed in low-alloying regions
than high-alloying regions. This can be discussed on the basis of the relation between
chemical composition and Ms temperature [18], which is given in Eq. ‎4-1. Fig. ‎4-11 shows the
influence of Mn and carbon segregation on Ms under two extreme conditions; (a) Mn
concentration changes in the range of 2.90-4.20 wt.%, as it was measured by EPMA
(Fig. ‎4-2b), and carbon concentration is constant (0.30 wt.%). (b) Carbon concentration
changes between 0.29-0.31 wt.%, based on ThermoCalc calculations in section ‎4.3.1, and Mn
concentration is as nominal composition (3.50 wt.%). According to Fig. ‎4-11, while carbon
segregation results in about 6 °C variations in Ms, Mn segregation leads to changes in Ms by
about 40 °C. Here, Si concentration is taken as the nominal concentration (1.6 wt.%).
Assuming carbon and Mn concentration as nominal composition of the steel, the measured
Si segregation leads to about 7 °C changes in Ms. Consequently, the microstructural banding
is mainly controlled by Mn segregation.
During the initial Q&P quenching, high-Mn regions, in which austenite is more stable than
in low-Mn regions, form lower fractions of M1. Therefore, austenite in high-Mn regions has
lower probability to receive sufficient carbon to remain at room temperature. Consequently,
high-Mn regions have higher fractions of unstable austenite that transform to M 2 during the
final quenching [25]. This explanation agrees well with Fig. ‎4-2a, in which higher fractions of
M2 are observed in high-Mn regions. It will be shown in section ‎4.4.2 that unstable austenite
may transform to bainite during isothermal holding. Accordingly, it is supposed that higher
fractions of bainite form in high-Mn regions.
4.4.2 Influence of carbon partitioning on bainite formation
All the Q&P specimens were isothermally held at the same temperature, therefore it is
expected that the kinetics of bainite formation be the same in all of them. However,
Fig. ‎4-8b shows that for a given isothermal holding time, the volume fraction of bainite is
higher in the specimens quenched to higher temperature.
63
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎4-11 The influence of Mn and carbon segregation on martensite start temperature.
This tendency could be related to the fact that specimens which were quenched to higher
temperatures contain a higher fraction of austenite during isothermal holding. This implies
that at constant isothermal holding time the normalised bainite fraction based on austenite
fraction, i.e. ratio of volume fraction of bainite (𝑓 𝑏 ) to volume fraction of austenite
(𝑓 𝑎 = 1 − 𝑓 𝑀1 ), be independent from quenching temperature. However, Fig. ‎4-12a shows
that the ratio of 𝑓 𝑏 to 𝑓 𝑎 depends on the initial quenching temperature and it is higher in
the specimens quenched to higher temperature. Therefore, higher fraction of austenite in
the specimens quenched to higher temperature is not the only reason of the increase of
bainite formation by the increase of quenching temperature.
An alternative reason for the increase of bainite fraction by increasing the quenching
temperature, is that specimens quenched to higher temperature contain higher fraction of
unstable austenite. This is based on the fact that at the beginning of isothermal holding,
before bainite formation, some regions of austenite become stable via carbon partitioning
process. Accordingly, austenite is divided to stable austenite, which does not decompose to
bainite and eventually is retained at room temperature, and unstable austenite, which may
decompose to bainite. This assumption implies that the plot of normalised bainite fraction
based on the unstable austenite fraction (the ratio of 𝑓 𝑏 to the volume fraction of unstable
austenite (𝑓 𝑈𝑛𝑆𝐴 )) versus isothermal holding time be independent from the quenching
temperature. Here, 𝑓 𝑈𝑛𝑆𝐴 = 1 − 𝑓 𝑀1 − 𝑓 𝑆𝐴 and 𝑓 𝑆𝐴 is volume fraction of stable austenite
and it is assumed as 𝑓 𝑅𝐴 after 5 s of isothermal holding. This assumption is based on the fact
that there is a negligible bainite formation during this interval. Fig. ‎4-12a shows that the
ratio of 𝑓 𝐵 to 𝑓 𝑈𝑛𝑆𝐴 for the specimens with different quenching temperature are close to
each other. This shows that higher fraction of bainite in the specimen quenched with higher
temperature is related to the higher fraction of unstable austenite in these specimens.
The initial interfaces between martensite and unstable austenite act as potential sites of
bainite nucleation [26]. However, it might be expected that the carbon partitioning leads to
formation of a thin film of stable austenite, RA, in martensite-unstable austenite interfaces
and therefore no bainite nucleates in these regions. This can be examined by observation of
martensite-unstable austenite interfaces in EBSD micrograph of the specimen QT220-5 in
Fig. ‎4-5. Note that here M1-M2 interfaces are considered as martensite-unstable austenite
interfaces, since M2 regions were unstable austenite during the isothermal holding.
64
Chapter 4
Fig. ‎4-12 (a) The normalised bainite fraction based on the volume fraction of austenite (𝑓 𝑎 = 1 − 𝑓𝑀1 ) as well
as unstable austenite fraction (𝑓 𝑈𝑛𝑆𝐴 = 1 − 𝑓 𝑀1 − 𝑓 𝑆𝐴 ) and (b) the kinetics of bainite formation (solid line)
compared to the experimental data points (circles) for the specimens QT220, QT240 and QT260.
As can be seen, there are M1-M2 interfaces that are free from RA and can be potential
place of bainite nucleation. The kinetics of bainite formation during isothermal holding of
the Q&P process can be simulated on the basis of two points. First, the overall kinetics is
controlled by the nucleation in austenite-austenite interfaces, bainite-austenite interfaces as
well as martensite-unstable austenite interfaces. Second, stable austenite does not
decompose to bainite. Then, the kinetics of bainite formation is expressed based on the
model developed in [27] as:
𝑑𝑓 𝐵
𝑑𝑡
= (1 − 𝑓 𝐵 − 𝑓 𝑀1 − 𝑓 𝑅𝐴 )(1 + 𝜆𝑀1 𝑓 𝑀1 + 𝜆𝐵 𝑓 𝐵 )𝜅,
‎4-9
in which 𝜆𝐵 and 𝜆𝑀1 are autocatalytic parameters of bainite nucleation at bainite-unstable
austenite interfaces and martensite-unstable austenite interfaces, respectively. κ is a
temperature dependent rate parameter and is given by [28]:
𝑍𝛿
−𝐾1 Ί
𝜅 = 𝜈 𝐷 𝛼𝑚́ 𝑒𝑥𝑝 (
𝐴
𝑅
−𝑄
) (𝑇ℎ − 𝑇)𝑒𝑥 𝑝 ( 𝑅𝑇𝑏),
‎4-10
where 𝜈 is the attempt frequency, 𝛿 is the effective thickness of the austenite grain
boundary, 𝑍 is a geometrical factor, αḿ is a kinetic parameter describing the rate of
martensite formation and DA is the parent austenite grain diameter. The parameter 𝐾1 is a
material constant and Ί =
respect to temperature.
d(∆𝐺𝑚 )
d𝑇
, i.e. the derivative of the maximum driving force ∆𝐺𝑚 with
65
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
In Eq. ‎4-10, 𝐾1 Ί (kJmol-1K-1) is given by (169.8 kJmol-1 -𝑄𝑏 )/(705K) and 𝑄𝑏 (kJmol-1) can be
calculated from [28]:
𝑄𝑏 = 89 𝑋𝐶𝑈𝑛𝑆𝐴 + 10𝑋𝑀𝑛 + 12𝑋𝑆𝑖 + 2𝑋𝐶𝑟 + 1𝑋𝑁𝑖 + 29𝑋𝑀𝑜 ,
‎4-11
𝑀
𝑋𝐶𝑈𝑛𝑆𝐴 (wt.%) is the carbon concentration in unstable austenite and is assumed as the 𝑋𝐶 2
after 5 s of isothermal holding. The temperature Th (°C) is the highest temperature at which
a displacive transformation can occur and is given by [28]:
𝑇ℎ = 835 − 198 𝑋𝐶𝑈𝑛𝑆𝐴 − 91𝑋𝑀𝑛 − 15𝑋𝑆𝑖 − 73𝑋𝐶𝑟 − 36𝑋𝑁𝑖 − 87𝑋𝑀𝑜,
‎4-12
The model is applied to simulate bainite formation in the specimen QT260-200. In the
present study, 𝜈=1013 s-1, 𝑍=6, 𝛿=1 nm and 𝛼ḿ = 0.015 K-1 [28]. 𝐷𝐴 is estimated as 5 μm from
𝑀
SEM observation of the specimen QT260-5. Given that 𝑋𝐶𝑈𝑛𝑆𝐴 and 𝑓 𝑆𝐴 are equal to 𝑋𝐶 2 and
𝑓 𝑀2 , these parameters are estimated 0.45 wt.% and 0.06, respectively. Note that to simplify
the calculation Mn and Si concentrations are assumed as the nominal composition. The
autocatalysis coefficients are chosen 13 and 6 for austenite-martensite and austenite-bainite
interfaces, respectively, for an optimum agreement between the model and the
experimental fraction-time curve, as shown in Fig. ‎4-12b. Bainite formation during the
isothermal holding of the specimens QT220 and QT240 is simulated under the assumption
that 𝜈, 𝑍, 𝛿, 𝛼𝑚 , 𝐷𝐴 , 𝜆𝐵 and 𝜆𝑀1 are independent of the quenching temperature. Fig.12b
shows that the model successfully simulates bainite formation in the specimens QT220 and
QT240. For the specimens QT180 and QT200, in which most of austenite becomes stable
after 5 s of isothermal holding, the simulations predict no bainite formation. This explains
the insignificant fraction of bainite in these specimens.
4.4.3 Influence of carbon partitioning, bainite formation and carbide precipitation on the
microstructure development
Presence of carbides in the SEM micrograph of the as-quenched specimen in Fig. ‎4-3a is
due to auto-tempering of martensite during quenching. This process has been reported for
steels with high Ms temperature [29]. Therefore, it can be claimed that ε-carbides in M1, as
detected in the TEM analysis of the specimen QT180-200 in Fig. ‎4-4c, are formed during the
initial quenching. According to ThermoCalc calculations, ɛ-carbide is unstable at isothermal
holding temperature (400 °C). Therefore, ɛ-carbide re-dissolves during this step and provides
carbon for equilibrium cementite formation or for further carbon partitioning to austenite.
Due to the slow kinetics of cementite formation in this high Si steel and fast kinetics of
carbon partitioning from martensite[2], most of carbon partitions to austenite. Carbon
depletion of martensite induces further decomposition of ɛ-carbide[30]. On the other hand,
carbon partitioning simulations, in section ‎4.3.3, show that all carbon in martensite diffuses
to austenite within 1 s of isothermal holding. Note that the simulations assume that carbon
in martensite is in solid solution. Therefore, carbon that remains in the microstructure after
1 s of isothermal holding can be assumed to be precipitated as ε-carbide. The gradual
decrease in carbon content of M1 as a result of longer isothermal holding, that is observed in
Fig. ‎4-10, is due to the decomposition of carbides and subsequent carbon partitioning.
Moreover, the carbon content of M1 is higher in the specimens quenched to lower
temperatures. The reason is that specimens quenched to lower temperatures have a higher
fraction of M1 and therefore a higher amount of carbon precipitates as carbide.
66
Chapter 4
Furthermore, the quenching process is longer in the specimens which were quenched to
lower temperatures and therefore carbon has more time to precipitate.
Simulations of carbon partitioning predicts that 200 °C is the optimum quenching
temperature. As can be seen in Fig. ‎4-7a, the influence of isothermal holding on the
calculated volume fractions of RA (𝑓 𝑅𝐴 ) depends on whether the quenching temperature is
above or below the optimum temperature. This agrees well with the calculations done in
[31]. Increasing the isothermal holding time from 5 s to 200 s does not influence the
simulated 𝑓 𝑅𝐴 of the specimens quenched to the temperature equal or below the optimum
temperature (specimens QT180 and QT200). In view of the small austenite grain size and
regarding the fact that carbon partitioning is sufficient to stabilize all the austenite, austenite
becomes stable within 5 s of isothermal holding and further tempering does not influence
the microstructure. This is exemplified in Fig. ‎4-6a for the specimen QT180. For the
specimens quenched to temperatures above the optimum temperature (specimens QT220,
QT240 and QT260), the calculated 𝑓 𝑅𝐴 decreases by increasing the isothermal holding time
from 5 to 200 s. During 5 s to 200 s of isothermal holding, carbon that was initially
accommodated near martensite-austenite boundaries becomes almost homogeneous inside
the austenite grains. Due to the fact that the total amount of carbon is not sufficient to
stabilize all austenite, homogenization of carbon inside austenite decreases the fraction of
stable austenite. The fact that after 5 s of isothermal holding the measured 𝑓 𝑅𝐴 of
specimens QT180, QT200 and QT220 is lower than the calculated ones can be related to the
considerable degree of carbide precipitation in M1 (Fig. ‎4-10). Keeping the isothermal
holding conditions constant, the calculated and the measured 𝑓 𝑅𝐴 of the specimens QT240
and QT260 are similar. This is due the fact that in these specimens a small fraction of carbon
precipitated as carbide, see Fig. ‎4-10.
Increasing the isothermal holding time to 50 s increases the measured 𝑓 𝑅𝐴 of the
specimens QT180 and QT200 to values close to the simulated ones. This can be explained by
the decomposition of some of ε-carbides and consequently carbon partitioning from M 1 to
austenite. This agrees well with Fig. ‎4-10 which shows that 𝐶 𝑀1 decreases during 50 s of
isothermal holding. Decomposition of ε-carbide and carbon partitioning from martensite
during 50 s of isothermal holding also explains the increase of 𝐶 𝑅𝐴 (Fig. ‎4-7c) and the
decrease of 𝑓 𝑀2 (Fig. ‎4-9a) in the specimens QT180 and QT200. Isothermal holding longer
than 50 s does not significantly change the microstructure properties of the specimens
QT180 and QT200.
At isothermal holding times shorter than 5 s, there is a limited fraction of bainite in the
specimens QT220, QT240 and QT260 (Fig. ‎4-8b). Therefore, in this time interval carbon
partitioning from martensite to austenite controls the microstructure development. During
isothermal holding longer than 5 s, carbide free-bainite forms in these specimens which is
accompanied by carbon diffusion from bainitic ferrite to austenite. The steel contains a high
concentration of Si and therefore it is supposed that carbide free-bainite is formed during
isothermal holding. In this regard, carbon diffusion from bainitic ferrite to austenite as well
as carbon diffusion from martensite to austenite are responsible for the increase of
measured 𝑓 𝑅𝐴 (Fig. ‎4-7a) to the values higher than calculated ones, moving the carbon
distribution of M2 toward higher concentration (Fig. ‎4-9b), the increase of 𝐶 𝑅𝐴 of the
specimens (Fig. ‎4-7c) and the decrease of volume fraction of M2 (Fig. ‎4-9a). However,
Fig. ‎4-10 shows that during 5 s to 200 s of isothermal holding the amount of carbon
partitioning from martensite is not significant and therefore carbon diffusion from bainitic
67
Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
ferrite is the prominent mechanism. It is worth emphasizng that in the current study bainite
forms only from unstable austenite. In this matter, bainite formation decreases the fraction
of unstable austenite (M2) and does not decrease the fraction of stable austenite.
4.5
Conclusions
The interaction between carbon partitioning, carbide precipitation and carbide-free
bainite formation is studied during the application of the Q&P process to a 0.3C-1.6Si-3.5Mn
(wt.%) steel with non-homogenous chemical composition. The main conclusions are:
68

Precipitation of ɛ-carbides during the first quenching reduces the amount of
carbon in solid solution in martensite. Therefore the amount of carbon
partitioning to the austenite is lower than according to simulations of carbon
partitioning. Regarding the fact that the partitioning of carbon present in carbides
requires the decomposition of the carbides and in view of slow kinetics of carbide
decomposition, full completion of the carbon partitioning process can be achieved
after isothermal holding time longer than predicted by simulations of carbon
partitioning.

At the initial stage of isothermal holding, carbon partitioning stabilizes a certain
fraction of austenite. This stable austenite does not decompose to bainite during
the isothermal holding and is retained at room temperature. In the specimens
with higher quenching temperature, carbon partitioning stabilizes a lower fraction
of austenite and therefore a higher fraction of bainite is formed.

Bainite formation reduces the volume fraction of secondary martensite, formed
from unstable austenite, by two mechanisms. First, bainite formation is
accompanied by carbon diffusion from bainite to austenite. This results in
stabilization of a part of the unstable austenite. Second, bainite forms from
unstable austenite and consequently decreases the fraction of unstable austenite.
Chapter 4
4.6
References
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austenite after martensite transformation”, Acta Mater. vol. 51, pp. 2611-2622, 2003.
[2] D. V. Edmonds, K. He, F. C. Rizzo, B. C. De Cooman, D. K. Matlock and J. G. Speer,
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vol. 438–440, pp. 25–34, 2006.
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2014.
[4] J. G. Speer, F. C. Rizzo, D. K. Matlock and D. V. Edmonds, “The quenching and partitioning
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Materialia, 50, 697–9 (2004)”, Scr. Mater., vol. 52, pp. 83-85, 2005.
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migration and carbon diffusion during annealing of martensite–austenite microstructures in
steels”, Scr. Mater., vol. 59, pp. 159-162, 2008.
[7] M. J. Santofimia, L. Zhao and J. Sietsma, “Microstructural evolution of a low-carbon steel
during application of quenching and partitioning heat treatments after partial
austenitization”, Metall. Mater. Trans. A, vol. 40, pp. 46-57, 2009.
[8] H. K. D. H. Bhadeshia and D. V. Edmonds, “The bainite transformation in a silicon steel”,
Metall. Trans. A, vol. 10, pp. 895-907, 1979.
[9] E. J. Seo, L. Cho and B. C. De Cooman, “Application of quenching and partitioning (Q&P)
processing to press hardening steel”, Matal. Mater. Trans. A , vol. 45, pp. 4022-4037, 2014.
[10] J. Mola and B. C. De Cooman, “Quenching and Partitioning (Q&P) processing of
martensitic stainless steels”, Metal. Mater. Trans. A, vol. 44, pp. 946-967, 2013.
[11] Y. Toji, G. Miyamoto and D. Raabe, “Carbon partitioning during quenching and
partitioning heat treatment accompanied by carbide precipitation”, Acta Mater., vol. 86, pp.
137–147, 2015.
[12] J. G. Speer, A. M. Streicher, D. K. Matlock, F. C. Rizzo and G. Krauss, “Quenching and
partitioning: A fundamentally new process to create high strength trip sheet
microstructures”, in Materials Science and Technology Meeting, Chicago, IL; United States,
2003.
[13] S. Murphy and A. Whiteman, “The precipitation of epsilon-carbide in twinned
martensite”, Metall. Trans., vol. 1, pp. 843-848, 1970.
[14] M. J. Santofimia, L. Zhao, R. Petrov, C. Kwakernaak, W. Sloof and J. Sietsma,
“Microstructural development during the quenching and partitioning process in a newly
designed low-carbon steel”, Acta Mater., vol. 59, pp. 6059–6068, 2011.
[15] M. J. Santofimia, R. Petrov, L. Zhao and J. Sietsma, “Microstructural analysis of
martensite constituents in quenching and partitioning steels”, Mater. Charact., vol. 92, pp.
91–95, 2014.
[16] T. Swarr and G. Krauss, “The effect of structure on the deformation of as-quenched and
tempered martensite in an Fe-0.2 pct C alloy”, Metall. Trans. A, vol. 7, pp. 41-48, 1976.
[17] A. J. Clarke, Ph.D. Thesis, Colorado School of Mines, pp. 31-32, 2006.
[18] S. M. C. van Bohemen, “Bainite and martensite start temperature calculated with
exponential carbon dependence”, Mater. Sci. Technol., vol. 28, pp. 487-495, 2012.
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Microstructural Characterization of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
[19] M. J. Duggin, “Thermally induced phase transformation in iron carbides”, Trans. Metall.
Soc. AIME, vol. 242, pp. 1091-1100, 1968.
[20] E. J. Fasiska and G. A. Jeffrey, “On the cementite structure”, Acta Cryst., vol. 19, pp. 463471, 1965.
[21] H. S. Yang and H. K. D. H. Bhadeshia, “Austenite grain size and the martensite-start
temperature”, Scr. Mater., vol. 60, pp. 493–495, 2009.
[22] M. J. Santofimia, L. Zhao and J. Sietsma, “Volume change associated to carbon
partitioning from martensite to austenite”, Mater. Sci. Forum, vols. 706-709, pp. 2290-95,
2012.
[23] D. P. Koistinen and R. E. Marburger, “A general equation prescribing the extent of the
austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels”,
Acta Metal., vol. 7, pp. 59-60, 1959.
[24] S. M. C. van Bohemen and J. Sietsma, “Martensite formation in partially and fully
austenitic plain carbon steels”, Matall. Mater. Trans. A, vol. 40, pp. 1059-1068, 2009.
[25] F. HajyAkbary, C. Kwakernaak, R. H. Petrov, G. Miyamoto, T. Furuhara, J. Sietsma and M.
J. Santofimia, “Effect of Mn segregation on the microstructure development of Q&P steels”,
in preparation.
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“Perspectives in high-strength steels: interactions between non-equilibrium phases”, in
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incomplete bainitic transformation in high-Si steels”, Int. J. Mat. Res., vol. 103, pp. 987-991,
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[28] S. M. C. van Bohemen, “Modelling start curves of bainite formation,” Metall. Mater.
Trans. A, vol. 41, pp. 285-296, 2010.
[29] G. R. Speich and W. C. Leslie, “Tempering of steel,” Metall. Trans. A, vol. 3, pp. 10431054, 1972.
[30] J. Gordine and I. Codd, “The influence of Si up to 1.5 wt.% on the tempering of a spring
steel,” J. Iron Steel Inst., vol. 207, pp. 461–467, 1969.
[31] A. J. Clarke, J. G. Speer, M. K. Miller, R. E. Hackenberg, D. V. Edmonds, D. K. Matlock, F.
C. Rizzo, K. D. Clarke and E. De Moor, “Carbon partitioning to austenite from martensite or
bainite during the quench and partition (Q&P) process: A critical assessment”, Acta Mater.,
vol. 56, pp. 16-22, 2008.
70
CHAPTER 5
5 Analysis of the Mechanical Behavior of
a 0.3C-1.6Si-3.5Mn (wt.%) Quenching
and Partitioning Steel*
Abstract.
Optimising the mechanical behavior of the Quenching and Partitioning (Q&P) steels
requires understanding of the relation between their microstructures and mechanical
properties. In this chapter, the yield strength of the constitutive phases of the Q&P
microstructures in a 0.3C-1.6Si-3.5Mn (wt.%) steel were analysed by applying physical
models and on the basis of data provided by detailed microstructure characterizations. The
yield strength of the Q&P microstructures were determined by considering the contributions
of the constituent phases on the yield strength on the basis of a composite law. The
experimentally measured and the calculated yield strengths are in the close agreement
which shows that the contributions of the phases on the yield strength depends linearly on
the yield strength and volume fraction of the phases. Initial martensite which has the highest
combination of the yield strength and volume fraction is the crucial factor in the yield
strength of the Q&P microstructures. Lowering the quenching temperature results in
increasing dislocation density of initial martensite and consequently increasing the yield
strength of initial martensite.
Keywords: Quenching and partitioning steels, Yield strength, Strengthening mechanism,
Dislocation density, martensite.
*
This chapter is based on a scientific paper:
F. HajyAkbary, J. Sietsma, G. Miyamoto, N. Kamikawa, R. H. Petrov, T. Furuhara, M. J. Santofimia , Analysis the
Mechanical Behavior of a Low Carbon Quenching and Partitioning Steels, in preparation.
71
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
5.1
Introduction
The "Quenching and Partitioning" (Q&P) process is known as a promising method for
developing steels with high strength and high ductility [1]. This process involves rapid
quenching of an austenitic microstructure to a temperature lower than the martensite-start
temperature (Ms) to form initial martensite (M1). The process is followed by an isothermal
treatment in which carbon partitions from supersaturated martensite to austenite and
stabilizes austenite sufficiently to be retained at room temperature [2]. During the
isothermal holding, bainite may form by decomposition of austenite, as it is discussed in
chapter 4 and [3]. The treatment is ended by quenching the microstructure to room
temperature. Secondary martensite (M2) may form during the final quenching, if some
regions of the austenite are not sufficiently stabilized, as it is shown in chapter 6 and [4].
Accordingly, the Q&P microstructures are composed of M1, bainite, M2 and retained
austenite (RA) and depending on the final microstructure, a varying range of yield strength
can be achieved. To tailor the desired yield strength, knowledge on the yield strength of the
individual phases as well as on the interaction of the phases on the final yield strength is
required.
The independent yield strength of the constituent phases are controlled by the lattice
defects in crystals [5]. Determining the yield strength of the phases requires the knowledge
of the lattice defects such as dislocations, the precipitates, the solute atoms and the grain
boundaries. However, owing to the complex assembling of phases in the Q&P steels the
quantitative analysis of the strengthening contributions of the phases is challenging. As an
example, three BCC phases (i. e. M1, bainite and M2) may co-exist in a Q&P microstructure
while their X-ray diffraction peaks overlap. Therefore, it is not possible to determine their
dislocation densities by using X-ray diffraction measurements. In this sense, most of the
researchers who study the relation between the microstructural and the mechanical
properties of the Q&P steels are mainly focused on the qualitative relation between the
fraction of the phases and the final mechanical properties of the Q&P steels [6-8].
In this study, the independent yield strengths of the constituent phases in the Q&P
microstructures in a 0.3C-1.6Si-3.5Mn (wt.%) steel are analysed on the basis of the
strengthening contributions from dislocations, grain boundaries, precipitates and solute
atoms. To do this, the microstructural properties of the individual phases are determined by
using dilatometry analysis, X-ray diffraction measurements, electron back scatter diffraction
(EBSD), transmission electron microscopy (TEM) and three-dimensional atom probe
tomography (3D-APT). The contributions of the phases on the yield strength of the Q&P
microstructures are determined by applying the composite law, with the aim of developing
an understanding of the key microstructural parameters that determine the yield strength of
the Q&P steels.
5.2
Theoretical calculation of the yield strength of the constituent phases
The yield strength of the Q&P microstructures can be estimated by considering the
contributions of the constituent phases on the basis of composite law:
𝜎𝑦𝑐 = ∑ 𝜎𝑦𝑖 𝑓 𝑖 ,
72
‎5-1
Chapter 5
in which 𝜎𝑦𝑐 is the calculated yield strength of the Q&P microstructure, 𝜎𝑦𝑖 is yield strength of
phase 𝑖 (𝑖=M1, bainite, M2 and RA) and 𝑓 𝑖 is the volume fraction.
5.2.1 Martensite and bainite
Yield strength of martensite (including M1 and M2) and bainite can be estimated as [9]:
2
2
𝑖
𝑖
𝑖
𝜎𝑦𝑖 = √(𝜎0 + 𝜎𝑠𝑠
+ 𝜎𝑔𝑏
) + (𝜎𝜌𝑖 + 𝜎𝑝𝑐𝑝𝑡
) ,
‎5-2
𝑖
𝑖
where 𝜎0 is the lattice friction stress for pure Fe, 𝜎𝑠𝑠
is the solid solution strengthening, 𝜎𝑔𝑏
𝑖
is the grain boundary strengthening, 𝜎𝜌𝑖 is the dislocation strengthening and 𝜎𝑝𝑐𝑝𝑡
is the
precipitate strengthening. Here, 𝑖 stands for M1, bainite or M2. Details on the calculation
procedure of the strengthening mechanisms are given in this section.

Solid solution strengthening
𝑖
The solid solution strengthening of each phase (𝜎𝑠𝑠
) is subdivided into contributions from
interstitial carbon atoms and from substitutional atoms:
𝑖
𝜎𝑠𝑠
= 𝜎𝐶𝑖 + 𝜎𝑠𝑡 ,
‎5-3
in which 𝜎𝐶𝑖 is the solid solution strengthening of carbon atoms and 𝜎𝑠𝑡 is the solid solution
strengthening from substitutional atoms. Note that the concentration of the substitutional
elements and therefore 𝜎𝑠𝑡 is the same in martensite and bainite, while 𝜎𝐶𝑖 depends on the
local carbon concentration. The solid solution strengthening from carbon atoms, in MPa, can
be calculated by [10]:
𝜎𝐶𝑖 = 1720𝑋𝐶𝑖
1/2
,
‎5-4
where 𝑋𝐶𝑖 is the concentration of carbon in solid solution in phase 𝑖, given in wt.%. The solid
solution strengthening from substitutional atoms, in MPa, is expressed as [11]:
𝜎𝑠𝑡 = 83𝑋𝑆𝑖 + 32𝑋𝑀𝑛 − 31𝑋𝐶𝑟 + 39𝑋𝐶𝑢 ,
‎5-5
where 𝑋𝑗 is the concentration of element j (j=Si, Mn, Cr and Cu), in wt.%. Eq. ‎5-4 and Eq. ‎5-5
were originally developed for steels with low alloying concentration (about 1-2 wt.%),
however, these equations have been applied in steels with alloying concentrations of about
3 wt.% [12].

Grain boundary strengthening
The effective grain sizes for bainite and martensite controlling the yield strength are
considered to be the bainite plate size [13] and the martensite block size [14], respectively.
𝑖
The grain boundary strengthening of each phase (𝜎𝑔𝑏
) is given by the Hall-Petch equation:
𝑖
𝜎𝑔𝑏
=
𝑘𝐻𝑃
𝑖
√𝑑𝑔𝑏
,
‎5-6
73
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
𝑖
in which 𝑑𝑔𝑏
is the effective grain size of martensite or bainite. 𝑘HP is the Hall-Petch slope
and is given as 0.21 MPa.m½ [14] for bainite and martensite.

Precipitate strengthening
𝑖
The precipitate strengthening, 𝜎𝑝𝑐𝑝𝑡
(MPa), in phase 𝑖 is expressed by [11]:
𝑖
𝜎𝑝𝑐𝑝𝑡
=(
𝑖
0.538𝐺𝑏√𝑓𝑝𝑐𝑝𝑡
𝑖
𝑑𝑝𝑐𝑝𝑡
)𝑙𝑛(
𝑖
𝑑𝑝𝑐𝑝𝑡
2𝑏
),
‎5-7
𝑖
𝑖
in which 𝑑𝑝𝑐𝑝𝑡
is the diameter of precipitates in nm in phase 𝑖, 𝑓𝑝𝑐𝑝𝑡
is the volume fraction
of precipitate, 𝐺 is the shear modulus of the material and 𝑏 is the Burgers vector. Here, 𝐺 =
80 GPa [15] and 𝑏= 0.248 nm [16]. This equation is valid for both ɛ-carbide and cementite
and has been applied for precipitates with diameter between 5-50 nm [11].

Dislocation strengthening
The contribution of dislocations to the yield strength of phase 𝑖, 𝜎𝜌𝑖 , can be approximated
from the well-known Taylor equation:
𝜎𝜌𝑖 = 𝛼𝑀𝐺𝑏√𝜌𝑖 ,
‎5-8
in which 𝛼 is a geometrical constant, 𝑀 is the Taylor factor and 𝜌𝑖 is the dislocation density
in phase 𝑖. Here, 𝑀 and 𝛼 are given as 2.73 and 0.25 [17], respectively.
5.2.2 Retained austenite
Young and Bhadeshia developed an empirical equation for estimating the yield strength
of austenite (𝜎𝑦𝑅𝐴 ), in MPa, as [13]:
𝑅𝐴
𝑅𝐴
𝑅𝐴
𝜎𝑦𝑅𝐴 = 15.4(4.4 + 23𝑋𝐶𝑅𝐴 + 1.3𝑋𝑆𝑖
+ 0.24𝑋𝐶𝑟
+ 0.94𝑋𝑀𝑜
+ 32𝑋𝑁𝑅𝐴 ),
‎5-9
where 𝑋𝑗𝑅𝐴 is concentration of element 𝑗 (𝑗 = C, Si, etc.) in RA, in wt.%. This equation has
been used to estimate the yield strength of austenite in low and medium carbon steels [13].
5.3
Experimental procedure
Cylindrical specimens with a length of 10 mm and a diameter of 3.5 mm were machined
parallel to the hot-rolling direction of 0.3C-1.6Si-3.5Mn (wt.%) steel sheets. The specimens
were austenitized at 900 °C for 180 s, quenched to 25 °C, 180 °C, 200 °C, 220 °C and 260 °C,
isothermally treated at 400 °C for 0 s, 5 s, 10 s, 50 s, 100 s and 200 s and finally quenched to
room temperature in a Bähr DIL 805 A/D dilatometer. In this chapter, the code QTxxx-y
identifies the specimen that was quenched to xxx °C and isothermally treated at 400 °C for y
seconds.
74
Chapter 5
Table ‎5-1 Dimensions of tensile specimens. The values are given in mm.
Specimen
T.3
T.4
T.10
T.120
Gauge length
3
4
10
120
Width
0.8
1
5
20
Thickness
1
1
1
1
After conventional metallographic preparation, specimens were etched with 2% Nital for
subsequent scanning electron microscopy (SEM) observations using a JEOL JSM-6500F field
emission gun scanning electron microscope (FEG-SEM) operating at 15 kV. The specimens
were metallographically prepared for electron backscatter diffraction (EBSD) examination
with a final polishing step of 0.05 μm using an OPS suspension for 1 hour. The EBSD analyses
were done by orientation imaging microscopy (OIM) on a FEI Nova 600 Nanolab dual-beam
(focused ion beam) FEG-SEM, under the following conditions: acceleration voltage 20 kV,
working distance 25 mm, tilt angle 70°, step size 50 nm. The orientation data were postprocessed with the TSL system. For calculating the grain size distribution only data with a
confidence index greater than 0.2 were analysed. Furthermore, selected specimens were
observed with a transmission electron microscope (TEM; Philips CM300) operated at 300 kV.
Thin-foil specimens of TEM analysis were prepared by twin-jet electropolishing at 25-30 V in
10% solution of HClO4 in ethanol at room temperature. Atom probe measurements were
carried out using LEAP 4000HR (CAMECA Co.) equipped in voltage pulse mode.
Three series of X-ray diffraction (XRD) measurements were performed;
(a) A continues scan XRD to determine volume fraction of RA (𝑓 𝑅𝐴 ) and the carbon
concentration of RA (𝑋𝐶𝑅𝐴 ). The analyses were performed by using a Bruker type D8-Advance
diffractometer, in a 2θ range from 30° to 135°, with Co Kα radiation. The reflections of {110},
{200}, {211} and {220} of the BCC structure and {111}, {200}, {220} and {311} of the FCC
structure were measured. The calculations of 𝑓 𝑅𝐴 and 𝑋𝐶𝑅𝐴 were performed in accordance
with the method given in [22].
(b) A stop scan XRD analysis to estimate dislocation density of martensite on the basis of
XRD peak broadening method. The XRD experiments were performed using a Bruker type
D8-Advance diffractometer in Bragg-Brentano geometry with graphite monochromator
equipped with a Bruker Vantec Position Sensitive Detector (PSD) using a Cu Kα anode.
Measurements were performed in the angular range of 40° (2θ) to 150° (2θ) to cover
reflections of {110}, {200}, {211}, {220} and {310) of the BCC structure with a step size of
0.02° (2θ) and a counting time of 10 s. More details about the of XRD analysis is given in [23].
(c) An in-situ XRD analysis to investigate the transformation of RA to martensite during
tensile deformation. The in-situ analysis was performed on flat tensile specimens which
were polished mechanically and electrolytically. Dimensions of the specimens were as T.3
specimens in Table ‎5-1. To do in-situ analysis, a Deben Microtest machine was mounted on
the Bruker D8 Discover machine with parallel beam geometry with Eulerian cradle equipped
with parallel sollerslit and graphite monochromator and using Co Kα radiation.
Measurements were performed in the angular range of 45° (2θ) to 107° (2θ), including the
reflections of {110}, {200}, {211} of the BCC structure and {111}, {200} and {220} of the FCC
structure, with a step size of 0.05° (2θ) and a counting time of 1 s. Note that the irradiated
surface area that was probed with the XRD machine is 4 mm × 3 mm. This area is larger
75
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-1 SEM micrograph of the specimens (a) QT25-0 and (b) QT180-5. M1 is initial martensite, RA is retained
austenite and M2 is secondary martensite.
than the flat surface of the tensile specimen, therefore it is required to correct the measured
integral area of peaks according to:
𝐴
𝑚
𝐴𝑐 = 𝑠𝑖𝑛𝜃
,
‎5-10
where 𝐴𝑚 and 𝐴𝑐 are the measured and the corrected integral area, respectively, and 𝜃 is
the Bragg angle.
Volume fractions of initial martensite (𝑓 𝑀1 ), formed during the initial quenching, bainite
(𝑓 𝐵 ), formed during the isothermal holding, and secondary martensite (𝑓 𝑀2 ), formed during
the final quenching, were calculated by applying the lever rule on the dilatometer data
during the initial quenching, isothermal holding and final quenching, respectively.
The tensile properties of the Q&P microstructures were measured by using miniature
tensile specimens. Two flat tensile specimens were prepared from dilatometric specimens.
Dimensions of the specimens are given as T.3 in Table ‎5-1. Since tensile properties were
measured by using sub-sized specimen, the influence of the specimen gauge length on the
tensile behaviour of steels was studied. This was doen by tensile testing of specimens with 3
mm, 4 mm, 10 mm and 120 mm from commercial dual phase (DP) and interstitial free (IF)
steels. Dimension of the specimens are given in Table ‎5-1. The tensile direction of the
specimen was parallel to the rolling direction of the sheets. The specimens T.3, T.4 and T.10
from DP and IF steels as well as the specimens QT180, QT200 and QT220 were tested using
Shimadzu AG-5000B tensile test machine while the tensile elongation was measured by a
CCD camera. The specimens T.120 from DP and IF steels were tested with a Schenk Trebel
tensile test machine and the strain was measured using a contact extensometer. The tensile
properties of the specimens QT25 were measured by using Deben Microtest machine while
the strain was measured by using crosshead displacement. The influence of the machine
compliance on the measured strain was corrected for by using the method developed in
chapter 2 and [18]. All of the tensile tests were done at an initial engineering strain rate of
2.1×10–3 s–1.
76
Chapter 5
Fig. ‎5-2 Volume fractions of retained austenite (RA), initial martensite (M 1), bainite (B), secondary martensite
(M2), ɛ-carbide and cementite in the specimens (a) QT25, (b) QT180, (c) QT200 and (d) QT220.
5.4
Results
5.4.1 Microstructure characterization

SEM observations
Fig. ‎5-1a and b show SEM micrographs of the specimens QT25-0 and QT180-5,
respectively. Fig. ‎5-1a shows that the specimen QT25-0 consists of martensite and carbides.
Microstructure of the specimen QT180-5 in Fig. ‎5-1b consists of internally etched features
(M1), blocky features (M2) and thin films of RA. The blocky features also included thin films of
RA. Precipitation of carbides inside M1 grains is evident. Similar microstructures were
observed in the other Q&P specimens, besides, the fraction of M2 decreases by reducing the
quenching temperature and increasing the isothermal holding time.

Volume fraction and carbon concentration of retained austenite
Volume fraction of RA (𝑓 𝑅𝐴 ) in the specimens QT25 is below the detection limit (0.02) of
XRD method and therefore it is assumed zero. Volume fraction of RA in the specimens
QT180, QT200 and QT220 are presented in Fig. ‎5-2b-d, respectively. Carbon concentrations
of RA in the specimens QT180, QT200 and QT220 were determined by using XRD analysis
and the results are shown in Fig. ‎5-3a.
77
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-3 Carbon concentration in (a) retained austenite (RA) and (b) secondary martensite (M2) in the
specimens QT180, QT200 and QT220.

Volume fraction and carbon concentration of secondary martensite
Volume fraction of M2 (𝑓 𝑀2 ) in the specimens QT180, QT200 and QT220 were
determined by applying the lever rule to the dilatometry results during the final quenching.
Considering that total fractions of the phases in some specimens is higher than unity and in
view of high uncertainty in estimation of f M2 , in these specimens, f M2 was calibrated by
balancing the total fraction to unity. High uncertainty in estimation of f M2 comes from the
fact that all of austenite does not transform to M2 during the final quenching, fitting a line to
the dilatometer curve and consequently determining the specimen dilatation is difficult.
Average difference between the normalised fraction of M 2 and the measured fraction of M2
is about 0.03. This value is considered as the uncertainty in the volume fractions of M2. For
the other phases the uncertainty of the volume fractions is considered 0.02. Fig. ‎5 2b-d
shows the normalised volume fraction of M2 in the specimens QT180, QT200 and QT220,
respectively. The carbon concentration of M2 was determined by fitting Koistinen and
Marburger (KM) equation [19] to the experimental plot of volume fraction of M 2 versus
temperature during the final quenching. A detailed description of this method is given in
section ‎4.3.6 and [3]. Fig. ‎5-3b illustrates the average carbon concentration of M2.

Volume fraction and carbon concentration of bainite
Volume fractions of bainite (f B ) were determined on the basis of the dilatometry data
during the isothermal holding. While there is an insignificant bainite formation in the
specimens QT180 and QT200, the volume fraction of bainite is considerable in the specimens
quenched to 220 °C. Volume fractions of bainite in the specimens quenched to 220 °C are
shown in Fig. ‎5-2d. Due to the low solubility of carbon in ferrite, the carbon concentration of
bainite is assumed zero.

Volume fraction and carbon content of initial martensite
Volume fractions of martensite in the specimens quenched to 25 °C were determined by
subtracting volume fractions of carbides from unity. The method to determine the volume
fractions of carbide is given in the following.
78
Chapter 5
Fig. ‎5-4 Carbon content of retained austenite (RA), initial martensite (M 1) and secondary martensite (M2) in
the specimens (a) QT180, (b) QT200 and (c) QT220.
Fig. ‎5 2a shows volume fractions of the phases in the specimens quenched to 25 °C.
Volume fractions of initial martensite (f^M1) in the specimens QT180, QT200 and QT220
were determined from dilatometry data and are shown in Fig. ‎5 2b-d, respectively.
The total carbon content of M1 can be determined by subtracting 𝐶 𝑅𝐴 and 𝐶 𝑀2 from the
total carbon content of the steel as:
𝐶𝑀1 = 𝐶 − 𝐶 𝑅𝐴 − 𝐶 𝑀2 ,
‎5-11
in 𝐶 𝑖 is carbon content of phase 𝑖 (𝑖 =M1, RA and M2) and 𝐶 refers to the nominal carbon
content of steel. The carbon content of each phase is given by:
𝐶 𝑖 = 𝑓 𝑗 𝑋𝐶𝑖 ,
‎5-12
where 𝑋𝐶𝑖 is carbon concentration of phase 𝑖. Note that the carbon content of bainitic ferrite
is assumed zero. Fig. ‎5-4a-c present the carbon content of RA and M2 in the specimens
QT180, QT200 and QT220, respectively. Inserting the nominal carbon content of the steel
(0.3 wt.%) in ‎5-11, 𝐶𝑀1 of these specimens were calculated and given in Fig. ‎5-4a-c.
79
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-5 Atomic maps for the distribution of carbon atoms in the specimens (a) QT180-5 and (b) QT180200.

Carbon in solid solution in initial martensite and secondary martensite
Note that the previous section determined the total carbon content of M 1. However,
calculation of the yield strength requires the knowledge of the fraction of carbon which is in
solid solution and in carbides. In order to determine the carbon concentration in solid
𝑀
solution in M1 (𝑋𝐶 1 ) in the specimens QT180-5 and QT180-200, several 3D-APT
measurements were performed in these specimens. Then, the average carbon concentration
in low carbon regions (regions with carbon concentration lower than nominal carbon
𝑀
concentration of the steel) was assumed as 𝑋𝐶 1 . This assumption was made based on the
fact that these specimens do not contain bainitic ferrite and M 1 is the only low-carbon
phase. Fig. ‎5-5a and 5b shows the distribution of carbon in M1 in the specimens QT180-5 and
QT180-200, respectively. Then, the average carbon concentration in the cylinders were
𝑀
considered as 𝑋𝐶 1 . However, it should be recognized that loss of Fe ions during the
detection leads to the overestimation of carbon concentration [20]. Then, the corrected
values of the carbon concentration with an uncertainty of 0.01 wt.%. is determined by using
the method given in [20]. The concentration of carbon in solid solution in M 1 in the
specimens QT180-5 and QT180-200 is estimated about 0.05±0.01wt.%. The low
concentration of carbon in solid solution in M1 agrees well with the simulations of carbon
partitioning process in chapter 3 and [3]. The simulations predict that almost all of carbon
atoms in solid solution partition to austenite during 1 s of isothermal holding at 400 °C and
therefore after isothermal holding longer than 1 s there is a small fraction of carbon in M1 as
solid solution. Regarding the high fraction of carbide in the SEM micrographs of the
specimens QT25, QT180, QT200 and QT220 and the view of the fact that simulation of the
carbon partitioning process predicts insignificant concentration of carbon in solid solution in
𝑀
M1, 𝑋𝐶 1 in the specimens QT25, QT180, QT200 and QT220 is considered 0.05±0.01wt.%.
The concentration of carbon in solid solution in M 2 can be considered as total carbon
concentration in M2. This is due to the fact that carbide does not precipitate in M2. More
details are given in next section.

Characterization of carbides
Fig. ‎5-6a shows bright field images of the specimen QT180-200 in transmission electron
microscopy. The carbide type was identified by selected area diffraction (SAD) analysis; here
only ε-carbide and cementite were considered. Fig. ‎5-6b illustrates the corresponding SAD
pattern and Fig. ‎5-6c shows the ideal SAD pattern corresponding to Fig. ‎5-6b.
80
Chapter 5
Table ‎5-2 Comparison between the calculated and experimentally obtained interplanar spacing of line 1 and 2
in Fig. 5-6 c. The 𝑑-spacing was calculated using the parameters 𝑎=2.752 Å, 𝑐=4.353 Å for ε –carbide [21] and
𝑎=4.525 Å, 𝑏= 5.900 Å, 𝑐=6.744 Å for θ [22].
Measured interplanar spacing (Å)
Calculated interplanar spacing (Å)
θ
ε
1
4.36
𝑑0001 =4.35
𝑑011 =4.06
2
2.40
𝑑11̅00 =2.38
𝑑200 =2.26
Fig. ‎5-6 Bright field micrograph of the specimens (a) QT180-200, (b) corresponding SAD pattern and (c) the key
diagram. Filled circles belong to ferrite reflections and open circles show carbide reflections and beam~//
[100]α~//[1120]ε.
Table ‎5-2 compares the calculated and the experimentally obtained interplanar spacing’s
between vector 1 and 2 in Fig. ‎5-6c. The measured interplanar spacing is close to the
literature values reported for ε-carbide. Additionally, the habit plane of the carbide is
parallel to the [220] direction of martensite which confirms the presence of ε-carbide in this
specimen. Presence of carbide in the specimen QT25 and the fact that formation of εcarbide has been often reported during quenching of low and medium carbon steels [25, 37]
indicate that ε-carbide is formed in martensite during the initial quenching of all specimens.
Cementite is not detected in the SAD pattern of the specimen QT180-200 which shows that
cementite does not form during the isothermal treatment. In view of the fact that
precipitate strengthening mechanism is the same for ɛ-carbide and cementite and
considering the slow kinetics of cementite precipitation during isothermal holding of this
steel at 400 °C in chapter 3 [23], the carbides in the specimens QT25, QT180, QT200 and
QT220 are taken as ɛ-carbide. Fig. ‎5-7a and 7b show dark field images of ε-carbide in M1 in
the specimens QT180-5 and QT180-200, respectively. The average diameter of ɛ-carbide
rods in these specimens was determined as 20±5 nm. Then, the average diameter of ɛcarbide in the specimens QT25, QT180, QT200 and QT220 is considered 20 nm and given in
Table ‎5-3.
According to TEM observations, the density of martensite grains with carbide precipitates
is lower in the specimens with higher quenching temperature. Considering that the volume
fraction of M1 decreases by increasing the quenching temperature, it can be concluded that
the presence of ε-carbide is not significant in M2. Volume fraction of ε-carbide (Fe3C) can be
estimated by subtracting the carbon concentration in solid solution from total carbon
81
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Table ‎5-3 Particle size of 𝜀-carbide (𝑑𝜀 ), block size of M1 for the Q&P specimens and martensite in the
specimens QT25 (𝑑𝑀 ), block size of M2 for the specimens Q&P (𝑑𝑀2 ) and plate size of bainite (𝑑𝐵 ).
Specimen
QT25
QT180, QT200 and QT220
𝒅𝜺 (𝐧𝐦)
20 ± 5
20 ± 5
𝒅𝑴 (µ𝐦)
0.4 ± 0.1
0.9 ± 0.10
𝒅𝑴𝟐 (µ𝐦)
-0.4 ± 0.10
𝒅𝑩 (µ𝐦)
-0.17 ± 0.10
Fig. ‎5-7 Dark field of ɛ-carbide in the specimens (a) QT180-5 and (b) QT180-200.
concentration of martensite. The volume fraction of carbide in the specimens QT25,
QT180, QT200 and QT220 are given in Fig. ‎5-2a-d, respectively.
Characterization of grain size of initial martensite, bainite and secondary martensite
Determination of block and plate size of M1, bainite and M2 requires identification of
these phases in the microstructures. The isothermal annealing results in decreasing the
dislocation density and carbon content of M1, therefore M1 and bainitic ferrite show similar
contrast in SEM and TEM observations, as well as similar Image Quality (IQ) in the EBSD
analysis. Therefore, it is not straightforward to distinguish M1 and bainite. Noted that the
thickness of bainite plates in silicon-rich alloys depends primarily on the strength of the
austenite at the transformation temperature and the chemical free energy change
accompanying transformation [23] which are both determined by the chemical composition
of the steel. Therefore, the grain size of bainitic ferrite (thickness of bainitic ferrite plate) is
considered equal to the thickness of bainitic ferrite plate in a steel with chemical
composition similar to the current steel (0.27C-1.98Si-2.18Mn wt.%) which was formed at
400 °C [23]. As it is recorded in Table ‎5-3, thickness of bainitic ferrite in the specimens QT220
is considered 0.17 µm. For each quenching temperature, the block sizes of M1 were
determined in the specimens containing a limited fraction of bainite, which is the case for
the specimens isothermally treated for only 5 s.
Fig. ‎5-8a shows a SEM micrograph of the specimen QT260-5. Although M1 and M2 can be
identified in the SEM micrograph as the etched and blocky features, respectively, their block
size should be characterized by using EBSD technique. Fig. ‎5-8b shows a combined grain
average Image Quality (IQ) and phase map of the specimen QT260-5, corresponding to the
region which is shown in Fig. ‎5-8a. In this analysis, first the EBSD measurements were
performed on the surface of the specimen.
82
Chapter 5
Fig. ‎5-8 (a) SEM micrograph of the specimen QT260-5, (b) combined IQ and phase map of the specimen QT2605 which includes the region shown in Fig. ‎5-8a. In Fig. ‎5-8b, black boundaries are high angle grain boundaries
(higher than 15°), green grains are RA and red grain are martensite, including M 1 and M2. (c) distribution of
grain average IQ of BCC phases in Fig. ‎5-8b in which yellow and blue columns belong to M1 and M2 grains and
the red column belong to M1 and M2 grains with similar IQ. (d) The combined IQ and phase map of the region
that is shown in Fig. ‎5-8a. In Fig. ‎5-8d, black boundaries are high angle grain boundaries (higher than 15°) and
green grains are RA, blue grains are M1 and yellow grains are M2. The red grains are M1 and M2 with similar IQ.
Subsequent SEM analysis requires cleaning from contamination by taking a fine polishing
step, small enough for the cleaning not to change the observed microstructure. In Fig. ‎5-8b,
RA grains are shown in green and martensite grains, including M1 and M2, are indicated in
red. In principle, M1 and M2 can be identified as regions with high and low grain-average IQ,
respectively. This is due to the fact that during the isothermal holding the carbon
concentration and dislocation density of M1 reduce to the values lower than M2 and this
implies that the IQ for M2 grains be lower than for M1 grains [24]. To identify the relation
between grain average IQ of M1 and M2, the grain average IQ of M1 and M2 grains in
Fig. ‎5-8a are considered in Fig. ‎5-8c. In Fig. ‎5-8c, blue and yellow columns show the IQ of M 1
and M2 grains, respectively. However, there are some of M1 and M2 grains which have
similar IQ and therefore their IQ overlap together. The IQ of such grains are shown with red
column in Fig. ‎5-8c. The corresponding grains IQ map of Fig. ‎5-8a is shown in Fig. ‎5-8d in
which blue grains are M1, yellow grains are M2 and green grains are RA. Fig. ‎5-8d also
contains some of red grains which are M1 and M2 grains with similar IQ. Accordingly, it is not
possible to define a threshold for IQ of M1 and M2 grains. However, M1 and M2 grains can be
determined as grains with high and low image quality (Fig. ‎5-8c), respectively, and this
method gives an estimation of M1 and M2 grain size.
83
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-9 Dislocation density in martensite in the specimens QT25 and initial martensite in the specimens
QT180, QT200 and QT220 as a function of (a) the quenching temperature and (b) isothermal holding time .
For the grains in which the normal direction of the specimen is not perpendicular to the
habit plane on the grain, the measured block size is the apparent block size. Therefore, it is
required to convert the apparent block size (𝑑́ 𝑖 ) to the true size (𝑑𝑖 ), perpendicular distance.
This was done based on the habit plane orientation of laths forming the block [14]:
𝑑𝑖 = 𝑑́ 𝑖 𝑠𝑖𝑛∅,
‎5-13
where ∅ is the angle between the normal of lath habit plane (∼(0 1 1)M//(1 1 1)A, M:
martensite, A: austenite) and the normal direction of specimen surface. The values of ∅
were measured from EBSD analysis. This approach is used to determine the block size of M1
and M2 in the specimens QT180-5 and QT220-5. For each specimen, the block sizes were
determined within five initial austenite grains. The block size of M1 and M2 in the specimens
QT180-5 and QT220-5 is 0.9±0.1 µm and 0.4±0.1 µm, respectively. Accordingly, similar block
size can be considered for M1 and M2 in the specimens QT200. The block size of martensite
in the specimens QT25 is determined from EBSD analysis of the specimen QT25-5 as 0.4±0.1
µm. Here, the block size of martensite is defined as the average size of grains with
misorientation higher than 15° with neighbouring grains [25]. Table ‎5-3 summarizes the
block size of M1 and M2 in the specimens QT180, QT200 and QT220 as well as martensite in
the specimens QT25.

Dislocation density of initial martensite, bainite and secondary martensite
The modified Williamson-Hall (MWH) approach and the modified Warren-Averbach
(MWA) approach have been widely used to determine the dislocation density of martensite
based on the XRD peak broadening [26]. However, these approaches are not robust and are
sensitive to the initial assumptions. A robust approach for characterization of dislocation
density has been developed in chapter 3 by combining the MWH and MWA approaches [27].
In this chapter, the mentioned method is applied to calculate the dislocation density.
Calculation of dislocation density of martensite in the specimens QT25 is straightforward,
since only one BCC phase is present in these microstructures.
84
Chapter 5
Fig. ‎5-10 Bright field images of twins in the specimens (a) QT180-5 and (b) QT180-200.
On the other hand, in the specimens QT180, QT200 and QT220, the XRD peaks of M1 and
M2 overlap together and therefore determining the dislocation density of these phases is
difficult. However, volume fractions of M2 are much smaller than of M1 and consequently
the full width at half maximum (FWHM) of the XRD peaks of M 1 is not strongly affected by
XRD peaks of M2. In this sense, dislocation density of M1 can be approximated by using the
XRD peak broadening analysis. Fig. ‎5-9a shows the dislocation density of martensite in the
specimens QT25 and dislocation density of M1 in the specimens QT180, QT200 and QT220.
The dislocation density of martensite is higher in the specimens with lower quenching
temperature. Moreover, Fig. ‎5-9b presents the influence of isothermal holding on the
dislocation density of martensite in the specimens QT25 and M 1 in the specimens QT180,
QT200 and QT220. The dislocation density of martensite decreases during the isothermal
holding, although the reduction of dislocation density is most pronounced during the first 5 s
of isothermal holding.
In this research, the dislocation density of M2 is assumed to be equal to the dislocation
density of the specimen QT25-0. The reason is that both M2 and martensite in the specimen
QT25-0 are formed during quenching to room temperature and they are not subjected to
the isothermal treatment. Presence of twins in M2 in the TEM micrograph of the specimens
QT180-5 and QT180-200 in Fig. ‎5-10a and Fig. ‎5-10b, respectively, agrees with the
observations made by Morito et al. [28]. According to that research, at high carbon
concentrations twins form in the martensitic microstructures.
The dislocation density of bainitic ferrite can be represented empirically as a function of
transformation temperature [29] by:
𝑙𝑜𝑔{𝜌𝐵 } = 9.3 +
6881
𝑇
−
1780300
𝑇2
,
‎5-14
in which 𝜌𝐵 is the dislocation density of bainitic ferrite, stated in m-2, and T is temperature in
K. This equation is based on experimental data over the temperature range 300-650 °C.
Considering the isothermal holding temperature as 400 °C, the dislocation density of bainitic
ferrite is approximated as 4 × 1015 m-2.
85
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-11 Engineering stress-strain curves (a) of miniature and standard specimens from DP and IF steels and
(b) of the specimens QT25-0, QT25-5, QT180-5 and QT220-5.
Fig. ‎5-12 Influence of the quenching temperature on the yield strength of specimens isothermally treated for
different times. The uncertainty in the yield strength measurement is about 40 MPa.
5.4.2 Yield strength measurements
Fig. ‎5-11a illustrates the engineering stress-strain curves of the miniature specimens (T.3,
T.4 and T.10) and standard specimens (T.120) for DP and IF steels. For both steels, the
specimen gauge length has insignificant influence on the yield strength and tensile strength.
This shows that specimen size does not affect the yield strength of the steels. Total
elongation of the specimens increases significantly by decreasing the specimen gauge
length. The effect of the specimen gauge length on the total elongation can be corrected by
applying the method given in chapter 2 and [30]. The engineering stress-strain curves of the
specimens QT25-5, QT180-5 and QT220-5 are illustrated in Fig. ‎5-11b. The yield strength of
the specimens were determined by using the 0.2% offset method.
Fig. ‎5-12 shows the yield strength of the specimens as a function of the quenching
temperature and for different partitioning times. The yield strength decreases by increasing
the quenching temperature, while the most pronounced decrease of the yield strength
occurs when the quenching temperature changes from 25 °C to 180 °C.
86
Chapter 5
Fig. ‎5-13 Influence of the applied stress on the volume fraction of retained austenite (RA)in the specimens
QT180-5 and QT180-200.
Fig. ‎5-14 Contribution of yield strength mechanisms from i. e. Peierls-Nabarro stress (𝜎° ), solid solution of
substitutional atoms (𝜎𝑠𝑡 ), solid solution of carbon atoms (𝜎𝐶 ), grain boundaries (𝜎𝑔𝑏 ), ɛ-carbide precipitates
𝜀
(𝜎𝑝𝑐𝑝𝑡
) and dislocations (𝜎𝜌 ) in initial martensite (M1), bainite (B) and secondary martensite (M2) in the
specimen QT220-200.
Fig. ‎5-12 also shows that for each quenching temperature, the isothermal process has
insignificant influence on the yield strength.
5.4.3 Stability of austenite
Analysing the influence of RA on the yield strength of the Q&P specimens requires
studying the stability of RA during the tensile tests. This was done by performing in-situ XRD
and microtensile analysis on the specimens QT180-5 and QT180-200. Fig. ‎5-13 shows the
influence of the applied stress on the volume fraction of RA. Before the yielding, the
reduction of volume fraction of RA is about 0.03. Therefore, induced martensite formation
has insignificant influence on the yield strength of the Q&P specimens.
87
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-15 Relation between calculated and measured yield strength of the specimens QT25,
QT180, QT200 and QT220.
5.5
Discussion
5.5.1 Calculation of contribution of strengthening mechanisms of martensite and bainite

Peierls-Nabarro stress
The Peierls-Nabarro stress is one of the strengthening mechanisms in Eq. ‎5-2. This is the
stress which is required to move a dislocation through a perfect lattice. A value of 41 MPa is
adopted by Speich and Swann [31] in their calculation for tempered martensite in a steel
containing 0.4 wt.% of carbon. In the current research, the Peierls-Nabarro stress in M1,
bainite and M2 is taken as 41 MPa, as is shown for the specimen QT220-200 in Fig. ‎5-14.

Solid solution strengthening
The solid solution strengthening from substitutional atoms (𝜎𝑠𝑡 ) is calculated by
substituting the chemical composition of the steel in Eq. ‎5-5. The concentrations of the
substitutional atoms are the same in all the specimens and the phases , therefore 𝜎𝑠𝑡 for M1,
bainite and M2 in the specimens QT180, QT200 and QT220 as well as for martensite in the
specimens QT25 is equal to 245 MPa. Fig. ‎5-14 shows that 𝜎𝑠𝑡 for M1, bainite and M2 in the
specimen QT220-200 is 245 MPa.
The concentration of carbon in solid solution in martensite in the specimens QT-25 and in
M1 in the specimens QT180, QT200 and QT220 is considered 0.05 wt.% on the basis of
3DATP measurements in Fig. ‎5-5. Then, Eq. ‎5-4 gives a value of 431 MPa for the solid
solution strengthening from carbon in M1 in the specimens QT180, QT200 and QT220.
Fig. ‎5-14 shows that solid solution strengthening from carbon in M1 in the specimen QT220200 is 431 MPa. The fact that no cementite was detected in TEM analysis of the specimens
QT180-200 indicates that isothermal holding has a neglectable influence on the solid
solution strengthening from carbon. The carbon concertation in bainite is assumed zero and
consequently the solid solution strengthening from carbon in bainite is zero.
88
Chapter 5
Fig. ‎5-16 The calculated yield strength of initial martensite (M1), bainite (B), secondary martensite (M2),
retained austenite (RA) as well as the calculated and measured yield strength of the specimens (a) QT180, (b)
QT200 and (c) QT220.
𝑀
The contribution of carbon solid solution strengthening in M 2 (𝜎𝐶 2 ) is calculated by
inserting the carbon concentration of M2 from Fig. ‎5-3b into Eq. ‎5-4. Fig. ‎5-14 shows that
𝑀
𝜎𝐶 2 in the specimen QT220-200 is 1040 MPa.

Grain boundary strengthening
The block size of M1 and M2 are determined as 0.9 µm and 0.4 µm (Table ‎5-3),
respectively, from EBSD analysis in section ‎5.4.1. Then, by applying Eq. ‎5-6, the contribution
𝑀1
of grain boundary strengthening of M1 (𝜎𝑔𝑏
) and the contribution of grain boundary
𝑀2
strengthening of M2 (𝜎𝑔𝑏 ) are calculated to be 221 MPa and 332 MPa, respectively.
Considering that bainitic ferrite plates have an average size of 0.17 µm, the contribution of
𝐵
grain boundary strengthening of bainite (𝜎𝑔𝑏
) is calculated as 509 MPa from Eq. ‎5-6. Note
that for the specimens QT180, QT200 and QT220 block and plate sizes of M1, bainite and M2
𝑀1
𝐵
𝑀2
are the same and consequently 𝜎𝑔𝑏
, 𝜎𝑔𝑏
and 𝜎𝑔𝑏
are the same. Fig. ‎5-14 represents the
𝑀1
𝑀2
𝐵
contributions of 𝜎𝑔𝑏 , 𝜎𝑔𝑏 and 𝜎𝑔𝑏 in the specimen QT220-200 which are 221 MPa, 332 MPa
and 509 MPa, respectively. According to Table ‎5-3, the block size of martensite in the
specimens QT-25 is 0.4 µm. The contribution of grain boundary strengthening in martensite
𝑀
(𝜎𝑔𝑏
) in these specimens is calculated from Eq. 5-5 as 332 MPa.
89
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
Fig. ‎5-17 Contributions of retained austenite (RA), to the yield strength of the specimens QT180,
QT200 and QT220.

Precipitate strengthening
TEM analysis of the specimen QT180-200 in Fig. ‎5-6 indicates the presence of ɛ-carbide in
M1. Then, the diameter of the carbide was determined by using dark field image of ɛ-carbide
in the specimens QT180-5 and QT180-200 (Fig. ‎5-7) about 20 nm. As indicated in Table ‎5-3,
the particle sizes of ε-carbide in the specimens QT25, QT200 and QT220 are also considered
20 nm. By substituting the fraction of ε-carbide in the specimens from Fig. ‎5-2 and the
average particle size of ε-carbide in Eq. ‎5-7, the contribution of the ε-carbide precipitate
strengthening is determined. The contribution of the ε-carbide precipitate in the specimen
QT220-200 is 405 MPa and it is given in Fig. ‎5-14.

Dislocation strengthening
According to Fig. ‎5-9a, the dislocation density of martensite decreases by increasing the
quenching temperature, however, it becomes more pronounced by increasing the
quenching temperature from 200 °C to 220 °C. A similar tendency of the increase of the
dislocation density with decreasing the quenching temperature was reported in [32], in
which the dislocation density of martensite was measured during quenching process by
using in-situ neutron analysis. In that analysis, the quenching process was interrupted and
then the dislocation density was measured. The reported influence of the quenching
temperature on the dislocation density is less pronounced than in the current research.
The reason is that at each quenching step and during the dislocation measurement,
martensite was tempered and some fraction of dislocations was annihilated. Therefore, the
measured dislocation density is lower than the actual dislocation density right after
quenching. The increase of dislocation density by decreasing the quenching temperature can
be explained by the fact that formation of new martensite crystallites induces
supplementary deformation (i.e. an increase in dislocation density) in the surrounding
martensite crystallites that have already formed. Furthermore, as the transformation
proceeds, the overall material deformation increases (in both martensite and austenite) and
then the material then becomes increasingly hard. Therefore, the shear stress associated
90
Chapter 5
with the formation of new martensite crystallites increases. This increases the dislocation
density of the newly formed martensite crystallite.
Fig. ‎5-9 shows that the dislocation density of M1 gradually decreases during isothermal
treatment, which is due to the tempering of martensite and annihilation of dislocations. The
dislocation strengthening in M1 is calculated by substituting the dislocation density of M1 in
Eq. ‎5-9. Fig. ‎5-14 shows the contribution of the dislocation strengthening in M1 in the
specimen QT220-200 is about 400 MPa.
The contribution of dislocation strengthening in M2 in all the specimen is calculated as
980 MPa by inserting the dislocation density of the specimen QT25-0 into Eq. ‎5-9.
Furthermore, the contribution of dislocation strengthening in bainite is estimated about 830
MPa by inserting the dislocation density of bainite (4× 1015 m−2 ) in Eq. ‎5-9. Fig. ‎5-14 shows
the contribution of the dislocation strengthening from M2 and bainite in the specimen
QT220-200 which are about 980 and 830 MPa.
5.5.2 Estimation of the yield strength of the constituent phases
Several equations have been used to determine the contributions of the strengthening
mechanisms to the yield strength of martensitic steel [9, 12]. Therefore, it is required to
evaluate the accuracy of Eq. ‎5-2 in the estimation of the yield strength of the martensitic
structure. In this matter, the yield strengths of the specimens QT25 are calculated by using
Eq. ‎5-2 and the results are compared with the measured yield strengths in Fig. ‎5-15. The
results show that Eq. ‎5-2 estimates the yield strength of martensitic structure with an
uncertainty of 5% which is an acceptable estimation in this study. Then, the yield strength of
M1, bainite and M2 in the specimens QT180, QT200 and QT220 are calculated by using
Eq. ‎5-2 and the results are shown in Fig. ‎5-16a-c, respectively. Secondary martensite, which
has the highest concentration of carbon in solid solution and dislocation density among the
other phases, shows the highest yield strength. Since the strengthening mechanisms of M 2
are almost similar in all the specimens, the yield strength of M2 is in the same range (17501900 MPa). The yield strengths of M1 in the specimens QT180 are in the range of 1190-1260
MPa while the yield strengths of M1 in the specimens QT220 change between 960 and 1060
MPa. This is mainly because of the reduction of the dislocation density of M 1 by increasing
the quenching temperature. Although there is a small degree of reduction in the dislocation
density of M1 during the isothermal holding it does not lead to significant change in the yield
strength of M1. The yield strengths of bainite in the specimens QT220 are calculated as 1121
MPa. Fig. ‎5-16c shows that the yield strength of bainite is independent of the isothermal
holding time. The yield strength of RA in the specimens QT180, QT200 and QT220 is
calculated by using Eq. ‎5-9 and the results are shown in Fig. ‎5-16a-c, respectively. The
calculated yield strengths of RA are about 120-140 MPa which are the lowest values among
other phases. The yield strengths of the Q&P specimens are also included in Fig. ‎5-16a-c. The
yield strengths of the specimens are lower than the yield strength of M 1, bainite and M2.
5.5.3 The yield strength of the Q&P specimens
The yield strengths of the specimens QT180, QT200 and QT220 are calculated by inserting
the volume fractions of the phases (Fig. ‎5-2) and the independent yield strengths of the
phases (Fig. ‎5-16) in Eq.‎5-1. The calculated yield strengths are compared with the measured
yield strengths in Fig. ‎5-16. There is a good agreement (about 95%) between the calculated
91
Analysis of the Mechanical Behavior of a 0.3C-1.6Si-3.5Mn (wt.%) Q and P Steel
and the measured yield strength. This shows that the contributions of the phases on the
yield strengths of the Q&P microstructures are linearly determined by the volume fractions
and the yield strength of the phases. The yield strength of the Q&P microstructures can be
analysed by considering the contributions of the phases on the yield strength, as it is shown
in Fig. ‎5-17. Due to the low yield strength of RA and fact that that there is an insignificant
induced martensite formation during the yielding region, RA has the lowest contributions on
the yield strength (Fig. ‎5-13). Moreover, M2 has a small contribution on the total yield
strength while it has the highest yield strength among the other phases. This can be
explained by considering the small volume fractions of M 2 in the microstructures. Regarding
to the small fractions of bainite in the specimens QT220, bainite has limited contributions on
the yield strength of these specimens. Due to the fact that M1 has the highest volume
fraction among the other phases and regarding high yield strength of M1, this phase has
highest contributions on the total yield strength. Therefore, M 1 is the most important phase
in tailoring the yield strength of the Q&P structures.
5.6
Conclusions
In this research the relation between the yield strength and microstructural properties of
the Q&P specimens is investigated and the following points are derived:
92

The contributions of the constituent phases on the yield strength of the Q&P
microstructures is determined by the multiplication of the independent yield
strength and volume fraction of the phases. Accordingly, the initial martensite
which has the highest volume fraction among the other microstructures and has
high yield strength has important effect on the yield strength of the Q&P
microstructures.

There is an inverse relation between the quenching temperature and dislocation
density of initial martensite. By increasing the quenching temperature the yield
strength as well as volume fraction of initial martensite decreases. Consequently,
the contribution of initial martensite on the yield strength of Q&P microstructures
decreases by increasing the quenching temperature.

Regarding to the low yield strength of retained austenite and in the view of the
fact that there is a limited degree of induced martensite formation prior to 0.2%
offset yield strength, retained austenite has insignificant influence on the yield
strength of the Q&P microstructures.
Chapter 5
5.7
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bainite,” Mater. Sci. Technol., vol. 6, pp. 592-603, 1990.
[30] F. HajyAkbary, M. J. Santofimia and J. Sietsma “Specimen size effects on the tensile
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94
CHAPTER 6
6 Optimising Mechanical Properties of a
0.3C-1.5Si-3.5Mn Quenching and
Partitioning Steel*
Abstract
This research is aimed to apply the Q&P process to a 0.3C-1.5Si-3.5Mn (wt.%) steel and to
develop a microstructure which has a good combination of tensile strength and ductility. In
this matter, different microstructures were developed during different Q&P processes and
their microstructural and mechanical properties were determined. Since, the mechanical
properties of the developed microstructures were measured by using microtensile tests, the
effect of the specimen size on the tensile properties is discussed and a correction procedure
is applied to convert the measured microtensile properties to the standard ones. The best
combination of tensile strength and ductility is achieved in the specimens with low
quenching temperature. Decreasing the quenching temperature, in one side increases the
volume fraction of initial martensite as well as the dislocation density of initial martensite
and thereby improves the tensile strength. On the other side, quenching to lower
temperature decreases the volume fraction of secondary martensite which has detrimental
effect on the ductility. A comparison with the measured mechanical properties of other
types of Advanced High Strength Steels (AHSS) shows the improved properties of the Q&P
steels.
Keywords: Quenching and Partitioning, Retained Austenite, Martensite, Microtensile test.
*
This chapter is based on a scientific paper:
F. HajyAkbary, M. J. Santofimia, J. Sietsma, Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenched
and Partitioned steel, Advanced Materials Research, 829 (2014) 100.
95
Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenched and Partitioned Steel
6.1
Introduction
An important concern of the auto-making industry is increasing safety and saving energy
[1]. For these purposes, Advanced High Strength Steels (AHSS) with high strength and
enhanced formability are being developed around the world. One of the most promising
heat treatments for creating a new generation of AHSS is the "quenching and partitioning"
(Q&P) process that produces a microstructure consisting of carbon-enriched films of
retained austenite between martensite laths [2]. The Q&P process involves full or partial
austenitization followed by a quench to a temperature lower than the martensite-start (Ms)
temperature in order to form a controlled fraction of martensite, which is called, in this
work, initial martensite (M1). The treatment is followed by an isothermal treatment, either
at or above the initial quenching temperature, named partitioning step. Within the
partitioning step, carbon partitions from supersaturated martensite to the austenite and
increases the stability of austenite to a level where it does not transform upon cooling to
room temperature and thus retained austenite (RA) is obtained. If some part of the
austenite does not become stable enough, it will transform to high-carbon martensite in the
final quenching. In this work, martensite formed in the final quenching is called second
martensite (M2). Different parameters such as quenching temperature, partitioning
temperature and time as well as the chemical composition of the steel influence the final
microstructure and consequently the mechanical properties of the Q&P steels [3].
The Q&P heat treatments are complicated and therefore the relations between
processing, microstructure and properties need to be studied accurately. An accurate
control of the heating process and temperature homogeneity within the specimen is
possible by heat-treating miniature specimens in the dilatometer. Resulting mechanical
properties can be evaluated by using the microtensile test. This method has been used for
mechanical properties characterization of Q&P steels [4]. Chapter 2 and [5] showed that not
only the microstructure but also the specimen dimensions affect the characterization of the
measured failure elongation of steels.
This research is aimed to apply the Q&P process to a 0.3C-1.5Si-3.5Mn (wt.%) steel and to
develop a microstructure which has higher tensile strength and ductility than other types of
AHSS. Different Q&P microstructures were created by quenching to a range of temperatures
and partitioning at 400 °C for 5 s. Mechanical properties of the Q&P microstructures were
measured by microtensile testing and the influence of the microstructure and specimen
dimensions on the measured tensile properties were discussed.
6.2
Experimental procedure
Cylindrical specimens with length of 10 mm and diameter of 3.5 mm were machined from
a 0.3 C-1.5 Si-3.5 Mn (wt.%) steel plate. Heat treatments were performed in a Bähr DIL 805
A/D dilatometer. The heat treatments included full austenitization at 900 °C for 180 s, rapid
quench (50 °C s-1) to 180 °C, 200 °C and 220 °C, partitioning at 400 °C for 5 s and finally
quenching to room temperature. In this research QTXXX denotes specimens which was
quenched to XXX °C.
96
Chapter 6
Table ‎6-1 Dimensions of miniature specimens (mm).
Specimen
Miniature
Width
0.8
Thickness
0.8
Gauge length
3.0
Grip area
3.5×3.0
Overall length
10.0
Fillet radius
0.5
Fig. ‎6-1 Dilatometry curves of the specimens (a) QT180, (b) QT200 and (c) QT220.
Two miniature tensile specimens were prepared from the middle of each cylinder and the
rest of the two curved parts were used for microstructure characterization. The dimensions
of the miniature tensile specimens are given in Table ‎6-1. The tensile tests were performed
by using Shimadzu AG-5000B tensile test machine while the tensile elongation was
measured by a CCD camera. Two miniature specimens were tested for each microstructure
and the average value of the tensile strength and elongation were considered. All tensile
tests were performed at an engineering (initial) strain rate of 2×10-3 s-1 at room temperature.
Uniform elongation is defined as the elongation at maximum stress.
Volume fractions and carbon contents of retained austenite were determined by X-ray
diffraction (XRD) experiments. XRD measurements were performed using a Bruker type D8Advance diffractometer equipped with a Bruker Vantec Position Sensitive Detector (PSD), in
the 2θ range from 30° to 135°, with Co Kα radiation in a step scanning at 0.05° with count
time of 2 seconds per point. The accuracy of the volume fraction of retained austenite (𝑓𝑅𝐴 )
and the carbon concentration of retained austenite are estimated as ± 0.02 and ± 0.05 wt.%,
respectively. Volume fractions of initial martensite (𝑓𝑀1 ) and second martensite (𝑓𝑀2 ) were
determined by applying the lever rule to the dilatometer data.
97
Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenched and Partitioned Steel
Fig. ‎6-2 (a) Volume fraction of phases and (b) carbon concentration of the retained austenite of the specimens
QT180, QT200 and QT220.
Since no significant expansion was observed in the dilatometry curves during the
partitioning step, the volume fraction of bainite was assumed zero. After conventional
metallographic preparation, specimens were etched with 2% Nital for subsequent SEM
observations using a JEOL JSM-6500F field emission gun scanning electron microscope (FEGSEM) operating at 15 kV.
6.3
Results and discussion
6.3.1 Microstructure Evolution during the Q&P process
Fig. ‎6-1a-c represent the dilatometry curves of the specimens QT180, QT200 and QT220
in the temperature range up to 410 °C. After full austenitization at 900 °C, the specimens
were quenched to 180 °C, 200 °C and 220 °C which led to formation of martensite. Here, the
martensite which is formed during the initial quenching is called initial martensite (M1).
Specimens did not show any dilatation at 400 °C. The specimen QT180 had a small dilatation
at the initial stage of the final cooling (indicated with a circle) due to specimen movement
when pumping helium during cooling. To examine the formation of M2, the initial part of the
dilatometer curve after partitioning step was extrapolated to room temperature (dashed line
in Fig. ‎6-1a-c) and any deviation of the relative length of the specimen from the extrapolated
line was attributed to the formation of M2. Secondary martensite was formed in all the
specimens and its volume fraction increases with increasing QT.
Volume fractions of the constituent phases are shown in Fig. ‎6-2a. For all the
microstructures, 𝑓𝑀1 decreases with increasing QT, which is in agreement with the KoistinenMarburger equation [6]. Furthermore, 𝑓𝑅𝐴 and 𝑓𝑀2 increase by increasing the quenching
temperature. The influence of the quenching temperature on the carbon concentration of
RA is illustrated in Fig. ‎6-2b. Considering the high error bars in this measurements no clear
trend can be detected in variations of the carbon concentration of RA.
98
Chapter 6
Fig. ‎6-3 SEM micrographs of the (a) QT180, (b) QT200 and (c) QT220 microstructures.
Fig. ‎6-3a-c show SEM micrographs of the specimens QT180, QT200 and QT220,
respectively. The needle-like features are thin films of RA and the dark-etched areas are M1.
Big grains of M1 contain a considerable fraction of carbides. It seems that the carbon content
of the small grains of M1 partitioned to the surrounding austenite, since there is no visible
carbide precipitation inside them. By increasing QT, the grain size of M1 decreases and
therefore the volume fraction of the carbide decreases. There are some big blocks in all
microstructures, which could be attributed to M2 and RA. The volume fractions of the big
blocks increase with increasing QT. This is in agreement with Fig. ‎6-2a which shows
increasing of 𝑓𝑀2 with the quenching temperature.
6.3.2 Mechanical properties of the Q&P microstructures
Fig. 6-4a illustrates the engineering stress-strain curves of the specimens. Variations of
the yield strength and tensile strength of the Q&P specimens with quenching temperature
are shown in Fig. ‎6-4b. The error bar in this figure corresponds to the maximum scatter in
data. Tensile strengths of the specimens increases by increasing the quenching temperature.
On the other hand, the yield strength is lower in the specimens with higher quenching
temperature. This indicates higher TRIP effect during the plastic region in the specimens with
higher quenching temperature. The increase of the TRIP effect is related to higher volume
fractions of RA and lower stability of RA by increasing the quenching temperature.
99
Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenched and Partitioned Steel
Fig. ‎6-4 (a) Engineering stress-strain curves of the specimens and variation of (b) the yield and tensile strength
and (c) the uniform and total elongation and (d) tensile strength-elongation of the QT180, QT200, QT220
microstructures as well as tensile strength -elongation of other types of AHSS steels, e.g. DP600, DP1000 and
M1400 [5].
Fig. ‎6-4c shows that uniform elongation of the specimens increases as the quenching
temperature increases. It is known that the uniform elongation is mainly controlled by the
ductile phase which is RA in the Q&P steels. In this sense, higher 𝑓𝑅𝐴 in the specimens with
higher quenching temperature leads to higher uniform elongation. It should be recognized
that in addition to the volume fraction of RA, the morphology and carbon content of RA
influence the uniform elongation. However, in the current study the morphology and carbon
content of RA are almost similar in all microstructures and their effect on the uniform
elongation can be neglected. Fig. ‎6-4c also indicates that total elongations of the specimens
also increase with increasing the quenching temperature.
6.3.3 Q&P Microstructure with optimised mechanical properties
As it was discussed in chapter 2, The specimen gauge length has insignificant influence on
the yield strength and tensile strength of steels. Therefore the measured strength of the
Q&P specimens can be compared with the tensile strength of the specimens which were
measured by using standard specimens [5]. However, the total elongation of miniature
specimens is higher than standard specimens.
In view of this problem, a correction method was applied to correct for the effect of the
gauge length on the elongation [2]. For specimens having a length-to-width ratio of 4, as is
100
Chapter 6
the case of this work, the correction equation is expressed as it was given in chapter 2
(Eq.‎2-15) and [5]:
𝑒𝐶 = 0.55𝑒𝑀 ,
‎6-1
where 𝑒𝑀 is elongation of the miniature specimen and 𝑒𝐶 is the corrected elongation on
gauge length of Lₒ. Total elongations of the miniature specimens were corrected by using
Eq. ‎6-1. The corrected total elongation of the Q&P microstructures as a function of tensile
strength are shown in Fig. ‎6-4d. This figure also includes total elongation and tensile strength
of other types of AHSS, e.g. DP600, DP1000 and M1400, measured by A80 standard tensile
tests [5]. Among the developed Q&P microstructures, the combination of tensile strength
and elongation improves by increasing the quenching temperature and therefore the best
optimised tensile properties is achieved in the specimen QT220. The increase of the tensile
strength by decreasing the quenching temperature is due to the increase of TRIP effect.
Fig. ‎6-4d shows that tensile strength of the Q&P microstructures is in the range of fully
martensitic steels (M1400) while their total elongation is similar or even higher than DP1000
steel. The fine substructure within the Q&P microstructures is apparently responsible for the
improved strength and elongation levels. Also, this figure indicates that the Q&P steels could
fill a gap in this elongation/strength distribution.
6.4
Conclusion
Optimising mechanical properties of the Q&P steels requires good understanding of the
relations between microstructure and mechanical properties. Three series of
microstructures were developed by quenching the fully austenitic specimens to 180 °C
(QT180), 200 °C (QT200) and 220 °C (QT220) and partitioning at 400 °C for 5 s.
Microstructural evolution during the Q&P process was analysed and the influence of the
microstructure on the measured mechanical properties was discussed. The following results
were found:

The quenching temperature (QT) plays an important role on the stability of the
austenite and the formation of high carbon martensite (secondary martensite)
within final quenching. Increasing the QT from 180 °C to 220 °C leads to increasing
the volume fraction of second martensite from 4% to 17%.

While the uniform elongation increases with increasing the volume fraction of
retained austenite (ductile phase), the post-uniform elongation decreases with
increasing the volume fraction of secondary martensite (brittle phase).

The specimen QT180 shows a good combination of elongation and tensile
strength. This is due to the increase of the volume fraction of initial martensite as
well as the dislocation density of initial martensite by lowering the quenching
temperature. Furthermore, quenching to lower temperature reduces the volume
fraction of secondary martensite and consequently improves the ductility. The
developed microstructures can compete with other types of AHSS.
101
Optimising Mechanical Properties of a 0.3C-1.5Si-3.5Mn Quenched and Partitioned Steel
6.5
References
[1] M. Pfestorf, D. Copeland, “Great designs in steel seminar 2007”, American Iron and Steel
Institute”, www.autosteel.org, 2007.
[2] E. De Moor, P. J. Gibbs, J. G. Speer, D. K. Matlock, “Strategies for third-generation
advanced high-strength steel development”, Iron & Steel Technol., vol. 7, pp. 133-144, 2010.
[3] J. G. Speer, F. C. R. Assunção, D. K. Matlock, D. V. Edmonds, “The Quenching and
Partitioning, Process: Background and Recent Progress”, Mat. Res., vol. 8, pp. 417-423, 2005.
[4] H. Liu, X. Lu, X. Jin, H. Dong, J. Shi, “Enhanced mechanical properties of a hot stamped
advanced high-strength steel treated by quenching and partitioning process”, Scr. Mater.,
vol. 64, pp. 749-752, 2011.
[5] F. Hajy Akbary, M. J. Santofimia, J. Sietsma, “Specimen size effects on the tensile behavior
of various steels”, 2nd International Conference SSTT-Determination of Mechanical Properties
of Materials by Small Punch and Other Miniature Testing Techniques, Czech Republic, 2012.
[6] D. P. Koistinen and R. E. Marburger, “A general equation prescribing the extent of the
austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels”,
Acta Metall., vol. 7, pp. 59-60, 1959.
102
CHAPTER 7
7 Conclusions and recommendations
The primary research objective of this thesis was to analyse the relation between
microstructural and mechanical properties of the Quenching and Partitioning (Q&P) steels
with the aim of optimisation of their mechanical properties. This aim was followed by
developing various microstructures in a 0.3C-1.6Si-3.5Mn (wt.%) steel with non-homogenous
chemical composition, applying different Q&P conditions. In order to have an adequate
control of the heat treatment, the Q&P process was applied to sub-size specimens. The
presented research addresses the challenges of measuring the mechanical properties of
steels by using miniature tensile specimens. Moreover, new approaches have been
developed to analyse microstructural properties including volume fraction, carbon
concertation, dislocation density and grain size of the constituent phases.
103
Conclusions and recommendations
7.1
Conclusions
The main contributions of this research are structured in four major directions:
1.
A new approach has been developed to determine the elastic strain of miniature
specimens. Comparing tensile behaviour of miniature and standard specimens from
different grades of steels showed that yield strength and tensile strength as well as
uniform strain are not influenced by the specimen geometry. On the other hand, it
has been shown that using miniature specimens in tensile tests leads to an
overestimation of fracture strain, however, the measured strain can be corrected
by applying corrective equations.
2.
An improved method is developed to measure dislocation density of lath
martensitic steels by applying X-ray diffraction profile analysis. This was done by
combining the modified Williamson-Hall and modified Warren-Averbach methods.
The proposed method is independent of limitations due to the considered range of
the Fourier length.
3.
The relation between chemical composition, heat treatment parameters and
microstructural properties of Q&P steels is investigated. During the initial
quenching of an inhomogeneous material, larger fractions of initial martensite are
formed in Mn/C/Si-poor regions than in Mn/C/Si-rich regions. This leads to a nonhomogenous distribution of initial martensite in the matrix. Lowering the quenching
temperature, a higher fraction of the austenite transforms to initial martensite and
therefore microstructural banding decreases.
4.
Precipitation of ɛ-carbides during the first quenching reduces the concentration of
carbon in solid solution in martensite. Regarding the fact that the partitioning of
carbon present in carbides requires the decomposition of the carbides and in view
of slow kinetics of carbide decomposition, full completion of the carbon partitioning
process can be achieved only after isothermal holding times longer than predicted
by simulations of carbon partitioning. Moreover, a method was developed to
determine carbon concentration of secondary martensite (martensite that is
formed during final quenching) on the basis of dilatometry data.
5.
At the initial stage of isothermal holding, carbon partitioning stabilises a certain
fraction of austenite. This stable austenite does not decompose to bainite during
the isothermal holding and is retained at room temperature. Furthermore, bainite
formation reduces the volume fraction of secondary martensite, that is formed
from unstable austenite, by two mechanisms. First, bainite formation is
accompanied by carbon diffusion from bainite to austenite. This results in
stabilization of a part of the unstable austenite. Secondly, bainite forms from
unstable austenite and consequently decreases the fraction of unstable austenite.
6.
The yield strength of the constitutive phases of the Q&P microstructures were
analysed by applying physical models and on the basis of data provided by detailed
microstructure characterisations. The yield strength of the Q&P microstructures
were determined by considering the contributions of the constituent phases on the
yield strength on the basis of the composite law. The experimentally measured and
the calculated yield strength are in the close agreement, which shows that the
104
Chapter 7
___________________________________________________________________________
contributions of the phases on the yield strength depends linearly on the
independent yield strength and volume fraction of the phases. Accordingly, the
initial martensite, which has the highest volume fraction among the other
microstructures and has high yield strength, has important effect on in the yield
strength of the Q&P microstructures.
7.
7.2
There is an inverse relation between the quenching temperature and dislocation
density of initial martensite. By increasing the quenching temperature the yield
strength as well as volume fraction of initial martensite decreases. Consequently,
the contribution of initial martensite on the yield strength of Q&P microstructures
decreases by increasing the quenching temperature. Regarding to the low yield
strength of retained austenite and view of the fact that there is a limited degree of
induced martensite formation prior to 0.2% offset yield strength, retained austenite
has insignificant influence on the yield strength of the Q&P microstructures.
Recommendations for future research
In light of the research objectives sought in this thesis, the following suggestions are given
for future research:



In the current research, mechanical properties of constituent phases of the Q&P
microstructures are computed based on a physical model. In this matter, it is
suggested to experimentally determine the tensile behaviour of the constituent
phases.
Based on the fundamental knowledge developed on the relation between the
fraction of high carbon martensite and ductility, it is known that high carbon
martensite reduces the uniform and post uniform elongations. Analysing the crack
propagation and failure behaviour of the Q&P steels will provide the opportunity to
control the detrimental effect of high carbon martensite.
It was found that Mn segregation leads to formation of microstructural banding
during the Q&P process of high Mn steels. This means that final microstructure
contains big aggregates of high carbon martensite which facilitates the crack
propagation. Consequently, studying the influence of the microstructural banding on
the failure behaviour of Q&P steels is recommended.
105
Conclusions and recommendations
106
Summary
This Ph.D. thesis investigates the relation between microstructural and mechanical
properties of Advanced High Strength Steels (AHSS), with the goal of developing a
microstructure with optimised mechanical properties. Among different grades of AHSS,
Quenching and Partitioning (Q&P) steel which is composed of thin films of retained austenite
between carbon depleted martensite laths, is selected. Different Q&P microstructures were
developed in a 0.3C-1.6Si-3.5Mn (wt.%) steel with non-homogenous chemical composition.
Aiming an adequate control of the microstructural evolution during the Q&P process, the
heat treatments were performed on small specimens in the dilatometer. In view of this, the
thesis is divided into three parts: chapter 2 investigates the influence of specimen size on the
tensile behaviour of steels, chapters 3 and 4 outline the methods to characterize the
microstructural properties of the Q&P specimens and chapters 5 and 6 discuss the relation
between the mechanical and microstructural properties.
Chapter 2 studies the influence of the specimen geometry on the mechanical behaviour
of steels. Miniature and standard specimens from different grades of steels were tested in
tension. The results show that while the specimen geometry has insignificant influence on
the actual elastic strain of the materials, the elastic strain which is measured from the
crosshead displacements is higher than the actual strain. The reason is that the elongation of
the fillet zones and the machine compliance are recorded along with the specimen
elongation as the crosshead displacement. A mathematical model is developed to correct
the influence of the elastic strain of the fillet-zones and the machine compliance. For
different types of steels, the calculated elastic strain and the strain measured on the
standard specimens are in good agreement and consequently the proposed model can be
used for calculating the elastic strain of the miniature specimens from the crosshead
displacement. Moreover, it was found that the yield strength, ultimate tensile strength and
uniform elongation of steels are almost independent of the specimen gauge length. Total
elongation increases with decreasing the specimen gauge length. This is a result of the
calculation method, since the total elongation is calculated by dividing the elongation of the
specimen by the initial gauge length, which is smaller for miniature specimens. Since the
post-uniform elongation is independent of the specimen parallel zone, the measured total
elongation is higher in miniature specimens. A method was applied for converting the total
elongation of the miniature specimen to the total elongation obtained from standard ones.
In chapter 3 an improved method is developed to measure dislocation density of a lath
martensitic steel by applying X-ray diffraction profile analysis. This was done by combining
the modified Williamson-Hall equation (MWH) and modified Warren-Averbach (MWA)
methods. The proposed method is independent of limitations due to the considered range of
the Fourier length. This method leads to a dislocation density that is in good agreement with
the dislocation density determined based on the dislocation strengthening. The MWH
method, under the assumption of a fixed value for the dislocation distribution parameter,
was applied to calculate the dislocation density. The calculated dislocation densities are in
the range of the values determined from the dislocation strengthening. However, it was
found that the combined MWH and MWA method can be used as a quantitative method for
dislocation density calculations, with a better accuracy than just the MWH method.
107
Chapter 4 investigates microstructural development during application of the Q&P
process in a steel with inhomogeneous chemical composition. In place EPMA and SEM
analysed show that during the initial quenching, in Mn/C/Si-poor regions higher fractions of
initial martensite are formed than in Mn/C/Si-rich regions. This leads to a non-homogenous
distribution of initial martensite in the matrix. Lowering the quenching temperature, a
higher fraction of austenite transforms to initial martensite and therefore microstructural
banding decreases. Moreover, it was found that precipitation of ɛ-carbides during the first
quenching reduces the concentration of carbon in solid solution in martensite. Regarding the
fact that the partitioning of carbon present in carbides requires the decomposition of the
carbides and in view of slow kinetics of carbide decomposition, full completion of the carbon
partitioning process can be achieved only after isothermal holding times longer than
predicted by simulations of carbon partitioning. A method was developed to determine
carbon concentration of secondary martensite, martensite that is formed during final
quenching, on the basis of dilatometry data. Additionally, it was found that at the initial
stage of isothermal holding, carbon partitioning stabilizes a certain fraction of austenite. This
stable austenite does not decompose to bainite during the isothermal holding and is
retained at room temperature. In the specimens with higher quenching temperature, carbon
partitioning stabilizes a larger fraction of austenite and therefore a lower fraction of bainite
is formed. Furthermore, bainite formation reduces the volume fraction of secondary
martensite, formed from unstable austenite, by two mechanisms. First, bainite formation is
accompanied by carbon diffusion from bainite to austenite. This results in stabilization of a
part of the unstable austenite. Secondly, bainite forms from unstable austenite and
consequently decreases the fraction of unstable austenite.
Chapter 5 studies the relation between the yield strength and microstructural properties
of the constituent phases i.e. retained austenite, initial martensite, bainite and secondary
martensite. The in-situ X-ray diffraction analysis showed that there is an insignificant
austenite to martensite transformation prior as well as during yielding of steels. Therefore,
the induced martensite formation does not have significant influence on the yield strength.
Yield strength of initial martensite, bainite and secondary martensite which was estimated
by applying physical models are higher than the total yield strength of specimens. The
summation of the normalised yield strength of the constitute phases gives an acceptable
approximation of the total yield strength. In this matter, the reduction of the yield strength
of the Q&P specimens by increasing the quenching temperature could be related to the
decrease of the dislocation density of initial martensite.
Chapter 6 showed that a good combination of high strength and elongation is obtained by
decreasing the quenching temperature which provides a high fraction of initial martensite
with high dislocation density. Moreover, microstructures with high fraction of initial
martensite have higher fraction of retained austenite as well as low fraction of secondary
martensite, as a brittle phase, and therefore show high elongation. Mechanical properties of
the developed microstructures can compete with other types of AHSS steels.
108
Samenvatting
Dit proefschrift onderzoekt de relatie tussen de microstructurele en mechanische
eigenschappen van een aantal Advanced High Strength Steels (AHSS), met als doel het
ontwikkelen van een microstructuur met geoptimaliseerde mechanische eigenschappen. Uit
de verschillende kwaliteiten AHSS is gekozen voor Quenching en Partitioning (Q&P) staal, dat
is samengesteld uit dunne films van restausteniet tussen koolstofverarmd martensiet
lamellen. Verschillende Q&P microstructuren worden ontwikkeld in een staal met 0,3C-1,6Si3,5Mn (gew.%) met een niet-homogene chemische samenstelling. Bij het streven naar een
adequate controle van de microstructurele veranderingen tijdens een Q&P-proces, worden
warmtebehandelingen uitgevoerd op kleine proefstukken in een dilatometer. Met het oog
hierop wordt het proefschrift verdeeld in drie delen: hoofdstuk 2 onderzoekt de invloed van
grootte van het proefstukop het trekgedrag van het onderzochte staal. De hoofdstukken 3
en 4 beschrijven de methodes die de microstructurele eigenschappen van de Q&P
proefstukken karakteriseren en de hoofdstukken 5 en 6 bespreken de relatie tussen de
mechanische en microstructurele eigenschappen.
Hoofdstuk 2 bestudeert de invloed van de grootte/afmetingen/ geometrie van het
proefstuk op het mechanische gedrag van het onderzochte staal. Miniatuur- en
standaardproefstukken van verschillende soorten staal worden getest onder spanning. De
resultaten tonen dat, terwijl de geometrie van het proefstuk een te verwaarlozen invloed
heeft op de werkelijke elastische rek van het materiaal, de elastische rek die wordt gemeten
vanaf de kruiskopverplaatsingen hoger is dan de werkelijke rek. De reden is dat de
verlenging van de stripzones en de doorbuiging/compliantie van de machine samen met de
verlenging van het proefstuk wordt gemeten als de kruiskopverplaatsing. Een mathematisch
model is ontwikkeld om de invloed van de doorbuiging/compliantie van de machine op de
elastische rek van de stripzones te corrigeren. Voor verschillende soorten staal komen de
berekende elastische rek en de rek, gemeten op de standaardproefstukken, goed overeen en
bijgevolg kan het voorgestelde model worden gebruikt voor het berekenen van de elastische
rek in het miniatuurproefstuk uit de kruiskopverplaatsing. Bovendien blijkt dat de
vloeisterkte, maximale treksterkte en uniforme rek van staal bijna onafhankelijk van de
inspanlengte van het proefstuk zijn. De totale verlenging neemt toe bij afnemende
inspanlengte van het proefstuk. Dit is een gevolg van de berekeningsmethode, aangezien de
totale verlenging wordt berekend door de verlenging van het proefstuk te delen door de
oorspronkelijke inspanlengte, die kleiner is voor miniatuur proefstukken dan voor standaard
proefstukken. Aangezien de post-uniforme verlenging onafhankelijk is van de parallelzone
van het proefstuk is de gemeten totale verlenging hoger bij miniatuur proefstukken. De
methode werd gebruikt om de totale verlenging van het miniatuur proefstuk om te zetten in
de totale verlenging verkregen uit standaard modellen.
In hoofdstuk 4 wordt een verbeterde methode ontwikkeld voor de bepaling van de
dislocatiedichtheid van martensitisch staal door toepassing van röntgendiffractie
profielanalyse. Dit wordt gedaan door het combineren van de "modified Williamson-Hall"
vergelijking (MWH) en de "modified Warren-Averbach" (MWA) methode. De voorgestelde
werkwijze is onafhankelijk van de beperkingen als gevolg van het beschouwde bereik van de
Fourier-lengte. Deze methode leidt tot een dislocatiedichtheid die in goede
overeenstemming is met de dislocatiedichtheid, bepaald op basis van dislocatieversteviging.
Ook de gewone MWH methode, onder de veronderstelling van een vaste waarde van de
109
parameter voor de dislocatieverdeling, wordt toegepast om de dislocatiedichtheid te
berekenen. De zo verkregen dislocatiedichtheden liggen in het interval van waarden,
bepaald op basis van dislocatieversteviging. Er wordt vastgesteld dat de gecombineerde
MWH-MWA methode gebruikt kan worden als een kwantitatieve werkwijze voor
berekeningen van dislocatiedichtheden met een hogere nauwkeurigheid dan enkel de MWH
methode.
Hoofdstuk 4 onderzoekt de microstructurele veranderingen gedurende het Q&P-proces
bij een staal met een niet-homogene chemische samenstelling. EPMA analyse en SEM
beelden tonen aan dat tijdens het afschrikken in Mn/C/Si-arme gebieden grotere fracties van
het initiële martensiet gevormd worden dan in Mn/C/Si-rijke gebieden. Dit leidt tot een
inhomogene verdeling van initieel martensiet in de matrix. Door het verlagen van de
afschriktemperatuur transformeert een grotere fractie van austeniet naar initieel martensiet
en daardoor neemt het bandkarakter van de microstructuur af. Bovendien wordt vastgesteld
dat ontleding van ɛ-carbides tijdens de eerste keer afschrikken de concentratie koolstof in
vaste oplossing in martensiet vermindert. Omdat de (her)verdeling van de in de carbides
aanwezige koolstof de ontbinding van die carbides vereist en omdat de kinetiek van carbideontbinding langzaam verloopt, kan voltooiing van het koolstofverdelingsproces enkel bereikt
worden als de isotherme houdtijden langer worden genomen dan voorspeld door simulaties
van de koolstof-verdeling. Een werkwijze wordt ontwikkeld om, op basis van dilatometriedata, de koolstofconcentratie te bepalen van secundair martensiet, dat wordt gevormd
tijdens het laatste afschrikken. Bovendien wordt gevonden dat, in het aanvangsstadium van
isotherme warmtebehandeling, de koolstofverdeling een bepaald deel van het austeniet
stabiliseert. Dit stabiele austeniet ontbindt niet tot bainiet tijdens de isotherme
warmtebehandeling en het blijft behouden bij kamertemperatuur. In de proefstukken met
hogere afschriktemperatuur stabiliseert de koolstofverdeling een grotere fractie austeniet
en daardoor wordt een kleinere fractie bainiet gevormd. Bovendien vermindert
bainietvorming de volumefractie van secundair martensiet, gevormd uit instabiel austeniet,
door twee mechanismen. Ten eerste gaat bainietvorming gepaard met koolstoftransport van
bainiet naar austeniet. Dit resulteert in stabilisatie van een deel van het instabiele austeniet.
Ten tweede vormt bainiet uit instabiel austeniet en daardoor vermindert de fractie van
instabiel austeniet.
Hoofdstuk 5 bestudeert de relatie tussen de vloeigrens en microstructurele
eigenschappen van de samenstellende fases, namelijk rest-austeniet, initieel martensiet,
bainiet en secundair martensiet. De in-situ röntgen-diffractie-analyse toont aan dat er een
insignificante transformatie van austeniet naar martensiet optreedt, zowel voor als tijdens
het vloeien van het staal. Daarom heeft de geïnduceerde martensietvorming geen
substantiële invloed op de vloeigrens. De vloeigrenzen van initieel martensiet, bainiet en
secundair martensiet, die worden geschat door het toepassen van fysieke modellen, zijn
groter dan de totale vloeigrens van de proefstukken. Het totaal van de genormaliseerde
vloeigrens van de samengestelde fases geeft een aanvaardbare benadering van de totale
vloeigrens. Op dit vlak zou de verlaging van de vloeigrens van de Q&P-proefstukken, door
het verhogen van de afschrikkingstemperatuur, gerelateerd kunnen zijn aan de vermindering
van de dislocatiedichtheid van initieel martensiet.
Hoofdstuk 6 toont aan dat een goede combinatie van hoge sterkte en verlenging bereikt
wordt door verlaging van de afschriktemperatuur wat een hoge fractie initieel martensiet
110
met een hoge dislocatiedichtheid levert. Bovendien hebben microstructuren met een hoge
fractie initieel martensiet een hogere fractie rest-austeniet evenals een lage fractie van
secundair martensiet, een brosse fase, en tonen daardoor hoge verlenging. Mechanische
eigenschappen van de ontwikkelde microstructuren kunnen concurreren met andere types
van AHSS-staal.
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Acknowledgements
This research was carried out under the project number M41.10.11437 in the framework
of the Research Program of the Materials innovation institute (M2i) in the Netherlands
(www.m2i.nl). First of all, I would like to thank M2i for the financial support of this project.
My special thanks are dedicated to Dr. ir. Viktoria Savran, program manager of M2i, for her
support during my Ph.D. work.
Many thanks to my supervisors Dr. Maria J. Santofimia and Prof. dr. ir. Jilt Sietsma for
their tremendous commitment and great support during all these years. This work would not
have achieved significant progress without their effective supervision. Also, I would like to
express my appreciation to Prof. dr.ir. Leo A. I. Kestens, cluster leader of the project in M2i,
for useful discussions. Tata Steel company as the industrial partner of the project is thanked
for useful collaborations. The project benefitted significantly from discussions with Dr. Dave
Hanlon and Dr. Stefan van Bohemen, Tata Steel Research, Development and Technology.
Also the useful collaboration with Prof. dr. Roumen H. Petrov at Ghent University is
appreciated. I would like to thank Dr. Amarante J. Böttger at Delft University of Technology
for her useful discussions in X-ray diffraction analysis. I would like to sincerely appreciate
from Dr. Rob Delhez at Delft University of Technology for his help in X-ray diffraction analysis
and translation of the summary to Dutch. The special thanks goes to my colleagues and the
support staff at Materials Science Department of Delft University of Technology.
I would also like to express my sincere gratitude to Dr. Goro Miyamoto whose support
during the last year of my Ph.D. research helped me to finalize my thesis. I am grateful to
Prof. dr. Tadashi Furuhara for useful discussions and his hospitality during my stay in Tohoku
University, in Japan. I wish also express my great thanks to my colleagues in Tohoku
University for all of their support and efforts. I am also thankful to Dr. Naoya Kamikawa at
Hirosaki University, in Japan for his hospitality and support. Also, useful discussions with Dr.
S. Morito at Shimane University, in Japan are appreciated.
Finally, I would like to thank all of my friends for their love and support during my Ph.D
work. I would like to express my appreciation to my parents and my brother for their
support during this work. I wish to thank to my husband, Mohsen, for his endless support
and love. Last but not least, my special thanks goes to my little angel, Arshida, who was
penitently with me during this work.
Farideh HajyAkbary
Delft, June 2015
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List of publications
Journal papers:
1. F. HajyAkbary, J. Sietsma, G. Miyamoto, N. Kamikawa, R. H. Petrov, T. Furuhara and
M. J. Santofimia, Analysis of the Mechanical Behavior in the Quenching and
Partitioning Steels, submitted to Acta Materialia.
2. F. HajyAkbary, J. Sietsma, G. Miyamoto, R. H. Petrov, C. Kwakernaak, T. Furuhara and
M. J. Santofimia, On the relation between Mn Segregation and Microstructural
Banding in the Quenching and Partitioning Steels, to be submitted to Scripta
Materialia.
3. F. HajyAkbary, J. Sietsma, G. Miyamoto, T. Furuhara and M. J. Santofimia, Interaction
of Carbon Partitioning, Carbide Precipitation and Bainite Formation during the
Quenching &Partitioning Process in a Low Carbon Steel, submitted to Acta
Materialia.
4. F. HajyAkbary, J. Sietsma, A. J. Bӧttger and M. J. Santofimia, An Improved X-ray
Diffraction Analysis Method to Characterize Dislocation Density in Lath Martensitic
Structures, Material Science and Engineering A, vol. 639, pp. 208-218, 2015.
5. F. HajyAkbary, M. J. Santofimia and J. Sietsma, Elastic Strain Measurement of
Miniature Tensile Specimens, Experimental Mechanics, vol. 54, pp. 165-173, 2014.
6. F. HajyAkbary, M. J. Santofimia and J. Sietsma, Optimising Mechanical Properties of a
0.3C-1.5Si-3.5Mn Quenched and Partitioned Steel, Advanced Materials Research,
vol. 829, pp. 100-104, 2014.
7. F. HajyAkbary, M. J. Santofimia and J. Sietsma, Influence of the Partitioning
Treatment on the Mechanical Properties of a 0.3C-1.5Si-3.5Mn Q&P Steel, Advanced
Materials Research, vol. 922, pp. 224-229, 2014.
8. R. H. Petrov, F. HajyAkbary, F. R. Saz, J. Sidor, M. J. Santofimia and J. Sietsma, L.
Kestens,. Microstructure and Properties of Ultrafast Annealed High Strength Steel,
Materials Science Forum, vol. 753, pp. 554-558, 2013.
Conference presentations and proceedings:
1. F. HajyAkbary, J. Sietsma, G. Miyamoto, R. H. Petrov, M. J. Santofimia, Martensite
Variant Selection During Quenching and Partitioning Process of a Low Carbon Steel,
International Conference on Processing & Manufacturing Of Advanced Materials,
Processing, Fabrication, Properties, Application, Austria, 2016.
2. F. HajyAkbary, C. Kwakernaak, J. Sietsma and M. J. Santofimia, Effect of Mn and C
Segregation on the Microstructure Development of Q&P Steel, International
Conference on Solid-Solid Phase Transformation in Inorganic Materials, Canada,
2015.
3. B. Kim, F. HajyAkbary, J. Sietsma and M. J. Santofimia, On the Occurrence of Carbide
Precipitation during Quenching & Partitioning Heat Treatments, Materials Science &
Technology Conference, Germany, 2014.
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4. S. R. Pascual, F. HajyAkbary, A. Bojack, M. J. Santofimia and J. Sietsma, Effect of
Microstructure on the Mechanical Properties of a Quenched & Partitioned MediumCarbon, High-silicon Steel, National Student Conference in Metallic Materials,
United Kingdom, 2013.
5. F. HajyAkbary, M. J. Santofimia and J. Sietsma, Specimen Size Effects on the Tensile
Behavior of Various Steels, 2nd International Conference of Small Sample Test
Techniques, Czech Republic, 2012.
6. R. H. Petrov, F. HajyAkbary, F. R. Saz, J. Sidor, M. J. Santofimia, J. Sietsma and L.
Kestens, Ultra-fast annealing of HSLA steel, International Congress Materials and
Technologies, Bulgaria, 2012.
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About the author
Farideh HajyAkbary was born on March 21, 1983 in Qom, Iran. After graduating from
secondary school in 2002, she started the bachelor’s program in Materials Science and
Engineering at Iran University of Science and Technology in Tehran, Iran. She obtained her
B.Sc. degree in 2006 and followed her M.Sc. studies in Materials Characterization at
University of Tehran, Iran. Her M.Sc. research work, advised by Prof. dr. M. Nili-Ahmadabadi,
was focused on the microstructure evolution during equal channel processing of bainitic
steel. Considering publishing several articles in peer reviewed journal, her M.Sc. thesis was
selected as the distinguished M.Sc. thesis of University of Tehran. In July 2011, to develop
her academic experience in an international environment, she started her doctoral research
in Department of Materials Science and Engineering at Delft University of Technology, the
Netherlands. The title of her Ph.D. project was “Optimising mechanical behavior of new
advanced steels based on fine non-equilibrium microstructures”. The project was done
under the supervision of Prof. dr. ir. J. Sietsma and Dr. M. J. Santofimia. During her research,
she paid a research visit at Tohoku University, Japan, with Prof. dr. T. Furuhara and Dr. G.
Miyamoto from 1st Febraury-1st May 2015. Her motto is that ever since I lost hope I feel
much better.
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