Effect of the anisotropy of martensitic transformation on ferrite deformation in Dual-Phase steels

Abstract

The volume increase and shape change during austenite to martensite transformation in dual-phase (DP) steels are largely accommodated in the microstructure by the deformation of the surrounding ferrite matrix. Accurate estimation of transformation-induced deformation of ferrite via experiments and modeling is essential for predicting the subsequent mechanical behavior of DP steels. This study aims to illustrate the disadvantages of simplifying the anisotropic transformation deformation of martensite to isotropic dilatation for modeling the transformation-induced deformation of ferrite. A novel methodology is developed comprising sequential experimental and numerical research on DP steels to quantify transformation-induced strains in ferrite. This methodology combines the results of prior austenite grain reconstruction, phenomenological theory of martensite crystallography and electron backscatter diffraction (EBSD) orientation data to estimate variant-specific transformation deformation. Subsequently, by comparison of full-field micromechanical calculation results on a virtual DP steel microstructure with experimental EBSD kernel average misorientation and geometrically necessary dislocation measurement results it is shown that neglecting the shear deformation associated with the martensitic transformation leads to significant underestimation in the prediction of transformation-induced strains in ferrite.

Full text

Effect of the anisotropy of martensitic transformation on ferrite
deformation in Dual-Phase steels
Vibhor Atreya
a,
⇑
, Jan Steven Van Dokkum
a
, Cornelis Bos
a,b
, Maria J. Santofimia
a
a
Department of Materials Science and Engineering, Delft University of Technology, 2628 CD Delft, the Netherlands
b
Tata Steel, Research & Development, IJmuiden, the Netherlands
highlights

The proposed modeling approach
estimates martensitic
transformation-induced deformation
of ferrite in DP steels.

The anisotropy of transformations is
found to be a significant factor
determining the deformation of
ferrite.

An influence of PAG orientation and
the variant formation on
transformation-induced deformation
of ferrite is shown.

This study illustrates an essential step
towards a general macroscopic model
of plasticity in DP steels.
graphical abstract
article info
Article history:
Received 24 March 2022
Revised 18 May 2022
Accepted 29 May 2022
Available online 31 May 2022
Keywords:
Dual-phase steel
Martensitic transformation
Electron backscatter diffraction
Plastic deformation
Martensite variants
Micromechanical modeling
abstract
The volume increase and shape change during austenite to martensite transformation in dual-phase (DP)
steels are largely accommodated in the microstructure by the deformation of the surrounding ferrite
matrix. Accurate estimation of transformation-induced deformation of ferrite via experiments and mod-
eling is essential for predicting the subsequent mechanical behavior of DP steels. This study aims to illus-
trate the disadvantages of simplifying the anisotropic transformation deformation of martensite to
isotropic dilatation for modeling the transformation-induced deformation of ferrite. A novel methodology
is developed comprising sequential experimental and numerical research on DP steels to quantify
transformation-induced strains in ferrite. This methodology combines the results of prior austenite grain
reconstruction, phenomenological theory of martensite crystallography and electron backscatter diffrac-
tion (EBSD) orientation data to estimate variant-specific transformation deformation. Subsequently, by
comparison of full-field micromechanical calculation results on a virtual DP steel microstructure with
experimental EBSD kernel average misorientation and geometrically necessary dislocation measurement
results it is shown that neglecting the shear deformation associated with the martensitic transformation
leads to significant underestimation in the prediction of transformation-induced strains in ferrite.
Ó2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
The microstructure of dual-phase (DP) steels consists of hard
martensite embedded in a soft ferritic matrix, which provides DP
steels with a good combination of strength and ductility. As a
https://doi.org/10.1016/j.matdes.2022.110805
0264-1275/Ó2022 The Author(s). Published by Elsevier Ltd.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
⇑
Corresponding author.
E-mail address: V.Atreya@tudelft.nl (V. Atreya).
Materials & Design 219 (2022) 110805
Contents lists available at ScienceDirect
Materials & Design
journal homepage: www.elsevier.com/locate/matdes

result, DP steels find widespread use in the automobile sector [1–
3]. The martensite present in the microstructure of DP steels is typ-
ically formed when austenite present in the microstructure at
intercritical condition transforms into martensite upon rapid cool-
ing. The intercritical ferrite does not undergo any phase transfor-
mation during such an operation.
The austenite to martensite phase transformation is accompa-
nied by a volume increase and a shape change [4,5]. Macroscopi-
cally, this is observed as a unidirectional dilatation and shear
deformation of the transforming region. To accommodate the
newly formed martensite in the microstructure, the ferritic matrix
also undergoes deformation. The resulting local stresses in ferrite
exceed the elastic limit, which results in plastic flow and subse-
quent strain hardening [6,7].
The transformation-induced plastic deformation of ferrite gen-
erates a high density of unpinned dislocations near the ferrite/-
martensite interface [8–10]. Because of the high local stresses
already present in the ferrite matrix, the newly formed dislocations
start to move and interact even at a relatively low value of global
flow stress [11,12], which results in the typical characteristics of
DP steels such as a low elastic limit, continuous yielding, and a
high initial work hardening rate [13,14]. Moreover, the extent of
transformation-induced deformation of ferrite significantly influ-
ences the yield strength and hardening behavior of DP steels
[6,7], which highlights the need of estimating the
transformation-induced deformation of ferrite to understand the
global mechanical behavior of DP steels.
Until now, only indirect qualitative experimental studies have
been performed to estimate the transformation-induced deforma-
tion of ferrite. These include estimating the extent of ferrite defor-
mation by measuring the lattice misorientations by electron back-
scatter diffraction (EBSD) [1,15–17], and nanoindentation mea-
surements near the ferrite/martensite interface to measure the fer-
rite strain hardening due to martensitic transformation [1,16,18].
Direct measurement of transformation-induced deformation of
ferrite is challenging owing to the fact that martensitic transforma-
tion occurs at a very high rate due to its displacive nature.
Continuum mechanics based analytical models for
transformation-induced deformation of ferrite have been formu-
lated for regular shaped martensite islands [11,12]. While Sakaki
et al. [11] estimated the spatial extent of plastic deformation, Bour-
ell and Rizk [12] estimated the dislocation density increase in the
deformed ferrite matrix surrounding a spherical martensite island.
For microstructures consisting of martensite with more complex
shapes, the transformation-induced deformation of ferrite has
been described with a micromechanics based numerical modeling
approach [6,7,18]. The anisotropic character of the transformation
deformation, which comprises of shear and dilatation deformation
accompanying the formation of every martensitic variant from a
prior austenite grain (PAG), can be expected to have a strong influ-
ence on the transformation-induced deformation of ferrite. How-
ever, in the aforementioned works, the transformation
deformation was assumed to be comprised only of isotropic dilata-
tion of the prior austenite grains.
The assumption of isotropic dilatation implies that the shear
deformation associated with the formation of martensitic variants,
oriented differently in space, cancel each other out leaving only the
unidirectional dilatation part for consideration. The volume aver-
age deformation of all variants combined is then considered equiv-
alent to the isotropic dilatation of the PAG. This assumption is
reasonable when the prior austenite grain size is sufficiently large
to allow the formation of a maximum of twenty-four martensitic
variants with different orientations. However, in the case of DP
steels, the small prior austenite grain size (PAGS) allows only the
formation of a few martensitic variants [19–21]. Hence the volume
average shear of all variants is non-zero, and the volume average
deformation of all variants combined is less likely to be equal to
the isotropic dilatation of the PAG.
This study aims to show the disadvantages of simplifying the
anisotropic transformation deformation of martensite to isotropic
dilatation for studying the transformation-induced deformation
of ferrite in the case of relatively small PAGS, such as in DP steels.
Subsequently, quantification of the transformation-induced defor-
mation of ferrite is proposed using a novel methodology compris-
ing sequential experimental and numerical research on DP steels.
2. Approach and theoretical models
The approach to model the transformation-induced deforma-
tion of ferrite is schematically presented in Fig. 1. It is divided into
four steps and starts with an EBSD scan of the DP steel microstruc-
ture to obtain lattice orientations of ferrite and martensite, fol-
lowed by prior austenite grain reconstruction to estimate the
morphology and lattice orientation of prior austenite grains. The
anisotropic transformation deformation is then determined by
using the phenomenological theory of martensite crystallography
(PTMC) [5,22,23]. As per the authors’ knowledge, the application
of a crystallographic theory (such as the PTMC) to calculate the
magnitude of shear and dilatation components of transformation
deformation is missing in the previous modeling works concerning
the transformation-induced deformation of ferrite in DP steels.
Subsequently, the transformation is mimicked by subjecting all
reconstructed prior austenite grains, constrained by an initially
undeformed ferrite matrix, to the anisotropic deformation associ-
ated with the transformation. The resulting transformation-
induced deformation of ferrite is calculated using a micromechan-
ical model (Appendix A)[24]. The detailed explanation of every
step follows.
2.1. Analysis of the ferrite/martensite microstructure (Step 1)
The EBSD scan of a selected location within the DP steel speci-
men is carried out to obtain information about the distribution of
phases and crystal orientations. Although both ferrite and marten-
site are recognized in the EBSD as bcc crystal structures, martensite
regions exhibit low image quality in the EBSD scans. Therefore the
grain average image quality measure is used to distinguish marten-
site from ferrite regions [25]. The identified martensite data points
are used as input for the PAG reconstruction.
2.2. Prior austenite grain reconstruction (Step 2)
The most well-known orientation relation observed between
austenite and martensite is the Kurdjumov-Sachs (K-S) orientation
relationship (OR). However, the experimentally observed OR usu-
ally deviate from the ideal K-S OR [26]. Therefore it is necessary
to determine the observed OR in the specimen under study.
The misorientation matrix, M
ij
, between two martensitic vari-
ants, iand j, formed from the same prior austenite grain can be rep-
resented as [27,28]
M
ij
¼C
1
j
T
1
S
1
j
S
i
TC
i
;ð1Þ
where S
i
and S
j
are one of the 24 cubic symmetry matrices for the
prior austenite grains of i
th
and j
th
martensitic variants respectively,
C
i
and C
j
are one of the 24 cubic symmetry matrices for i
th
and j
th
martensitic variants as well, Tis the austenite-martensite orienta-
tion relationship matrix, and superscript 
1
denotes the inverse.
Here, the subscripts are not index notations. The term
T
1
S
1
j
S
i
T

results in 24 distinct rotations [27]. If the experimen-
tally observed misorientation matrix M
exp
, between the two
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
2

martensitic variants, iand j, is found to be close to one of the pre-
dicted 24
3
misorientation matrices according to Eq. 1, the two vari-
ants are assigned to the same PAG. However, prior to this
calculation, the OR matrix Tshould be ascertained. Manipulating
Eq. 1, we find that:
T¼T
1
S
1
j
S
i

1
C
j
M
exp
C
1
i
:ð2Þ
Since the OR matrix Tappears on both sides of Eq. 2, the equa-
tion can only be solved iteratively. Eq. 2then becomes:
T
nþ1
¼
T
1
n
S
1
j
S
i

1
C
j
M
exp
C
1
i
;ð3Þ
where nis the iterator, T
n¼0
is the initial guess for T
n
equal to the OR
matrix corresponding to the theoretical K-S OR, T
nþ1
is the OR
matrix calculated from M
exp
, and 
T
n
is the OR matrix corresponding
to average of all ORs determined in the previous iterations. The iter-
ative process is based on determining correct symmetry matrices by
using T¼
T
n
in Eq. 1and comparing all predicted values of misori-
entation matrix M
ij
with M
exp
. When there is no change in the cal-
culated symmetry matrices in subsequent iterations, the final OR
matrix 
T
n
is concluded to be the observed experimental orientation
relationship for the specimen under study.
The application of this reconstruction algorithm not only pre-
dicts the PAG shape but also helps in identifying the martensitic
variants and provides an estimate for PAG crystal orientation. This
information is further used to determine the anisotropic, variant-
specific transformation deformation.
2.3. Determination of martensitic transformation deformation (Step 3)
The austenite to martensite phase transformation comprises of
dilatation and shear deformation [5,22,29]. The transformation is
an invariant plane strain (IPS) deformation, where the invariant
plane is also called the habit plane. The dilatation part is perpen-
dicular to the invariant plane of martensitic transformation, while
Fig. 1. Illustration of the four steps involved in the calculation of transformation-induced deformation of ferrite. (a) The crystal orientations of ferrite and martensite phases
in the specimen are obtained from EBSD measurements (b) A PAG reconstruction algorithm provides the shape and the crystal orientation of the PAGs. (c) The anisotropic
transformation deformations are calculated using the phenomenological theory of martensite crystallography (PTMC) (d) The reconstructed PAGs are subjected to the
calculated anisotropic and variant-specific transformation deformations and the resulting transformation-induced deformation of ferrite is calculated via micromechanical
modeling.
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
3

the shear part is parallel to the invariant plane. According to the
phenomenological theory of martensite crystallography (PTMC),
the IPS can be expressed in the form of the following equation
[23,5,22]:
Z¼RBP;ð4Þ
where Zis the IPS shape deformation gradient matrix, Ris the rota-
tion matrix representing a rigid body rotation, Bis the bain strain
matrix, and Pis the lattice invariant shear matrix. The lattice invari-
ant shear is either in the form of slip, resulting in the formation of
laths in low carbon martensite, or in the form of twinning in high
carbon martensite [30]. Since martensitic transformation is an
invariant plane strain deformation, Zcan also be written in the fol-
lowing form [29]
Z¼Iþm^
d^
p;ð5Þ
where mis the magnitude of displacements in the direction of dis-
placement unit vector ^
d;^
pis the habit plane unit normal vector, and
is the outer product. The slip plane and direction for the lattice
invariant shear matrix Pare assumed to be f112g111
bcc
since it
is common to observe f112gstriations on the martensite surface
[31]. The shape deformation gradient matrix Zand the rotation
matrix Rare now calculated from Eqs. 4 and 5. The bain strain
matrix Bis calculated from lattice parameters of martensite and
prior austenite [29]. In this work, PTC lab software [32] is used to
obtain the solution for the shape deformation gradient matrix Z
by providing lattice parameters of austenite and martensite as
inputs.
Apart from the shape deformation gradient matrix Z, the PTMC
also enables the calculation of orientation transformation matrix T
using the PTC lab software [32]. The change in crystallographic ori-
entation due to martensitic transformation can be expressed in the
form of the following matrix equation [33,34]:
M¼TA;ð6Þ
where Mand Aare the orientation matrices consisting of three
orthogonal unit vectors representing the crystal orientation of
martensite and austenite respectively. The orientation transforma-
tion matrices of all 24 K-S variants of martensite are
T
k
¼C
k
TC
T
k
;ð7Þ
where Cis the cubic symmetry matrix, k=124, and 
T
is the
transpose of the matrix. Plugging Eq. 7into 6, we have
M
k
¼C
k
TC
T
k
A;ð8Þ
which gives the orientation matrices of all 24 variants formed from
a prior austenite grain. Similarly, the IPS solution Zfor all 24 vari-
ants formed from a single prior austenite grain is calculated using
the following equation:
Z
k
¼C
k
ZC
T
k
;ð9Þ
where Z
k
is the variant-specific deformation matrix which can only
be obtained if the correct symmetry matrix C
k
is identified. The
symmetry matrix C
k
which results in the minimum misorientation
angle between M
k
and the experimentally measured orientation
matrix of a variant is concluded to be the correct symmetry matrix
for that variant present at a specific martensite location in the EBSD
scan.
The deformation gradient matrix Z
k
is calculated with respect to
the reference frame of the prior austenite grain. Any given PAG has
an arbitrary orientation with respect to the specimen coordinates.
The correct IPS deformation gradient matrix for any martensitic
variant is [35]:
Z
kl
¼A
l
C
k
ZC
T
k

A
T
l
;ð10Þ
where A
l
is the orientation matrix of l
th
prior austenite grain. The
variant-specific deformation gradient matrices, Z
kl
, are used to cal-
culate the variant-specific eigenstrains 

kl
. Eigenstrains are the
strains developed in the material due to the inelastic processes such
as phase transformation and thermal expansion in the absence of
any external mechanical stress. The variant-specific eigenstrains


kl
are then used as input in micromechanical model to calculate
the transformation-induced strains in ferrite.
2.4. Micromechanical calculation of transformation-induced strains in
ferrite (Step 4)
In the last step of the proposed procedure, micromechanical cal-
culations are performed to estimate the transformation-induced
strains in ferrite matrix in response to austenite undergoing
variant-specific deformation while transforming into martensite.
A fast fourier transform based micromechanical model is used to
perform the calculations (Appendix A.1)[24]. The inputs for the
calculations are: a representative volume element (RVE) based on
a microstructural image of DP steel, a behavior law for each phase
involved in the RVE, the boundary conditions imposed on the RVE,
and the eigenstrains corresponding to martensitic transformation
deformation.
In the present micromechanical model calculations, the strains
are assumed to be infinitesimal. This means that the second-order
terms from the definition of finite strain are neglected. Therefore
any significant transformation-induced rotation in the material
contributes to the error in calculated transformation-induced
strains. A comparative analysis of the transformation-induced
deformation of ferrite is performed, in response to two different
assumptions regarding martensite transformation deformation.
The trends observed are considered representative given the
absence of local a priory known rotations. The intention here is
to use the simplest numerical method to elucidate the importance
of a key aspect of martensitic transformation.
3. Analysis of a DP steel microstructure using the proposed
method
3.1. Experimental procedure
A cold-rolled steel with composition Fe-0.14C-1.8Mn-0.24Si by
weight % and with an initial microstructure consisting of pearlite
and ferrite was cut into a specimen of dimensions
10 mm 4mm2 mm using electrical discharge machining.
Using a Bähr DIL A/D dilatometer, the specimen was heated at
5 K/s, kept at 1048 K for 5 min, and then quenched to obtain a
ferrite-martensite dual-phase microstructure. The specimen was
ground using SiC abrasive papers and subsequently polished using
3 and 1
l
m diamond paste. Further, it was electropolished using
Struers A2 electrolyte at 35 V, 277 K for 6 s. EBSD measurements
were made on a Zeiss Ultra 55 scanning electron microscope
(SEM) using the Edax Pegasus XM 4 Hikari EBSD system. The step
size of the scan was 50 nm. The EBSD scans were analyzed using
the TSL OIM version 7 software.
3.2. Application
3.2.1. Analysis of the ferrite/martensite microstructure
The microstructure of the specimen used in this work was
investigated in detail in a previous study [36].Fig. 2(a) shows the
image quality (IQ) map of the specimen obtained via EBSD scan.
The dark-colored martensite is present in the form of connected,
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
4

chain-like structures. The region used for PAG reconstruction and
RVE generation for micromechanical calculations is highlighted
in red and shown enlarged in Fig. 2(b). The RVE hence generated
is shown in Fig. 2(c) where blue and red colors represent ferrite
and martensite respectively.
The martensite volume fraction for the specific scanned location
of the specimen was calculated to be 0.56 [36], based on phase
identification using grain average image quality (GAIQ) measure-
ments [25,36]. GAIQ is the average image quality of all measure-
ment points within a grain. Using an angular tolerance value
between 0.5 and 1.5°to identify the grains followed by an applica-
tion of GAIQ measure enables excellent identification and quantifi-
cation of martensite in a DP steel microstructure [25,36]. The
equilibrium carbon content in ferrite at the intercritical tempera-
ture calculated using ThermoCalc software is 0.006 wt%. Conse-
quently, the carbon content in martensite was calculated to be
0.23 wt% [34] after the application of carbon mass balance. The lat-
tice parameters used in this study are a= 2.858 nm and
c= 2.885 nm for martensite and a= 3.565 nm for prior austenite
based on calculated carbon content [37].
The n
th
neighbor kernel average misorientation (KAM) at an
EBSD measurement point is the average misorientation of the n
th
nearest neighbor points with respect to that measurement point.
Since intra-grain misorientations arise due to dislocations, and
KAM values can be used to study the level of deformation within
individual grains [38].Fig. 2(d) shows the KAM maps of the region
of interest for the 5
th
nearest neighbors and a maximum 3°misori-
entation. As expected, the highest KAM values are observed for the
martensite regions. The KAM magnitude is higher in ferrite grains
near the phase boundary than in the grain interior. This becomes
apparent in Fig. 2(e), where the KAM values are grouped into five
uniformly spaced ranges and overlaid on the IQ map. The regions
near the phase boundary which show higher KAM are highlighted
in dashed white ellipses.
A local geometrically necessary dislocation (GND) density map
is calculated using MTex software [39–41], following the approach
given by Pantleon [42]. This approach involves the calculation of
dislocation density tensor at every measurement point of the EBSD
scan, whose components are determined using the lattice curva-
ture tensor. The lattice curvature tensor is calculated from the
misorientations between neighboring points in an EBSD scan.
Fig. 2(f) shows the result for the GND density calculations, with
some regions of relatively high GND densities highlighted with
black dashed ellipses.
3.2.2. Prior austenite grain reconstruction
To carry out the prior austenite grain (PAG) reconstruction, the
bcc ferrite phase data points in the EBSD maps were identified
using the grain average image quality (GAIQ) criterion and then
removed from the EBSD scan data. Reconstruction was then per-
Fig. 2. (a) Image quality (IQ) map of the specimen region scanned with EBSD (b) Specific area selected for this study (c) Generation of an RVE consisting of ferrite and
martensite (d) Kernel average misorientation (KAM) map for the selected region. KAM values are in degrees (e) KAM map for the selected region with values grouped into
specific ranges. Areas within white dashed white ellipses indicate higher KAM regions in ferrite (f) Geometrically necessary dislocations ðm
2
Þcalculated for the region of
interest. Some regions of higher GND density are highlighted with black dashed ellipses.
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
5

formed on the remaining martensite data points. Mtex version 5.6
toolbox available in Matlab [39] was used for carrying out the PAG
reconstruction.
Fig. 3(a) shows the reconstructed parent grain boundaries over-
laid on the IQ map of the region of interest shown in Fig. 2(b).
Figs. 3(b) and 3(c) show the packet and variant map of all the PAGs
shown in Fig. 3(a). The PAGs are very small and with a mean diam-
eter of 1.6
l
m and the average number of variants in a PAG is 6.
Fig. 4(a) shows the pole figure orientation of the largest prior
austenite grain, named ‘PAG 1’, indicated in Fig. 3(a). Fig. 4(b)
shows the pole figures of identified variants of ‘PAG 1’. The theoret-
ical pole figure of all 24 variants according to the OR identified dur-
ing the reconstruction is shown in Fig. 4(c). Experimental pole
figures of the variants match very well with the expected theoret-
ical pole figures. The experimental OR was found to be 1.1 degrees
off from theoretical K-S relationship.
A comparison of the pole figures of martensitic variants accord-
ing to theoretical K-S OR and those obtained via PTMC is shown in
Fig. 4(d). The misorientation between predictions of variant orien-
tation by K-S OR and that by PTMC is 3.22°. Neither the PTMC nor
the K-S OR predicts the experimentally observed ORs accurately.
As per the prediction of PTMC, the deviation between close-
packed planes of austenite (111)
c
and martensite (110)
a
decreases
from 0.4°to 0.3°when the carbon content increase from 0.1 wt% to
0.4 wt%. This is also corroborated in experiments where the
decrease in angular deviation for close-packed planes is from 0.8°
to 0.3°for the same increase in carbon content [43]. This shows
that even though PTMC may not capture the experimentally
observed OR, it can predict the changes in OR with varying carbon
content in austenite/martensite.
3.2.3. Determination of martensitic transformation deformation
The anisotropic transformation eigenstrain for martensite,

;aniso
kl
, for the k
th
variant of the l
th
austenite grain is calculated from
the anisotropic transformation deformation matrix, Z
kl
of Eq. 9
using the definition of Green-Lagrange strain (appendix A.1). The
value of min Eq. 5as calculated using PTMC theory is 0.26. The
transformation eigenstrain matrix for isotropic dilatation of
martensite,

;iso
, is of the form

;iso
¼ðD=3Þd
ij
, where Dis the vol-
umetric strain for austenite to martensite transformation and d
ij
is
the Kronecker delta representing the identity matrix. The volumet-
ric strain accompanying martensitic phase transformation depends
on the carbon content as [4,11]:
D¼
D
V
c
!
a
V
c

M
s
¼0:017 þ0:013X
C
;ð11Þ
where
D
V
c
!
a
is the change in specific volume due to phase transfor-
mation from austenite cðÞto martensite aðÞ;V
c
is the specific volume
of austenite, and X
C
is the carbon content of austenite in wt%.
For X
C
¼0:23;Dis calculated to be 2:010
2
. In Eq. 11, all transfor-
mation is assumed to occur at the martensite start temperature
(M
S
).
3.2.4. Micromechanical calculation of transformation-induced strains
in ferrite
Following PAG reconstruction, the micromechanical model cal-
culations are performed on the EBSD-based RVE to calculate
transformation-induced strains in ferrite. Depending upon the
variant identity, the corresponding eigenstrain is applied at the
locations of transformation. The lattice-invariant shear between
the martensitic laths is a part of the macroscopic shape deforma-
tion, and therefore does not need to be considered separately in
the micromechanical calculations. Assuming a fixed room-
temperature value of linear thermal expansion coefficient, the
thermal strain in ferrite due to quenching is in the order of 10
3
[44]. The strains induced in ferrite by anisotropic martensitic
transformations are expected to be higher in magnitude (in the
order of 10
1
) than the thermal strains, hence thermal strains are
not considered in this study.
The material behavior of martensite was assumed to be isotro-
pic elastic, while ferrite was considered isotropic elastic-plastic
with linear hardening in the following form:
r
0
¼
r
Y
þH
vM
;ð12Þ
where
r
0
is the flow stress,
r
Y
= 283 MPa is the uniaxial ferrite yield
strength, H= 595 MPa is the plastic modulus [24]-a parameter
which determines the level of strain hardening depending upon
strain magnitude and 
vM
is von-Mises equivalent strain. The values
of
r
Y
and Hwere chosen in such a way that the flow curve resulting
from Eq. 12 is the best linear fit for the experimentally observed
uniaxial tensile flow curve of ferrite with a similar carbon concen-
tration as shown in Fig. 5 [45]. The elastic modulus and Poisson’s
ratio for martensite (E = 203.5 GPa, m= 0.292) and ferrite
(E = 209 GPa, m= 0.289) are obtained from the literature [46].
For the EBSD based RVE, the spatial discretization in the x, y and
z directions is 332 291 1, equal to the number of measurement
Fig. 3. (a) Reconstructed prior austenite grain boundaries in black overlaid on image quality map of the region of interest (b) Variants of every PAG coloured differently, with
a maximum of 24 variants per PAG (c) Packets of every PAG coloured differently, with a maximum of 4 packets per PAG.
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
6

Fig. 4. (a) Pole figure of the largest prior austenite grain (PAG 1) indicated in Fig. 3(a); (b) The pole figures of identified variants. (c) The theoretical pole figure of all 24
variants according to the OR identified during the reconstruction (d) A comparison of the pole figures of martensitic variants according to theoretical K-S OR and those
obtained via PTMC.
Fig. 5. (a) Linear approximation of ferrite hardening curve (in red dotted line) based on experimental flow curves of ferrite presented in reference [45].
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
7

points in the area selected to study from the EBSD scan. Periodic
boundary conditions are applied in the plane of the 2D surface.
No information is available on grain morphology out of the plane.
Therefore, a mechanical limit is applied and the average far-field
strain orthogonal to the 2D surface is set to zero, i.e., plane strain
condition was considered for the in-plane direction. The conver-
gence of the numerical iterative process was evaluated by compar-
ison with the required accuracy of 10
5
m
1
.
Fig. 6 shows the von-Mises equivalent map of (a)
transformation-induced strains in ferrite due to isotropic transfor-
mation eigenstrains,

iso;tr-ind
vM
, and (b) transformation-induced
strains in ferrite due to anisotropic transformation eigenstrains,

aniso;tr-ind
vM
. The magnitude of deformation at all locations in ferrite
is found to be higher in Fig. 6(b) than in 6(a). The high strain
regions highlighted with black dashed circles at the top and bot-
tom edge of both Figs. 6(a) and 6(b), arise because of the imperfect
periodicity of the microstructural region selected for this study. It
is to be noted that the elastic lattice strains within martensite are
numerically obtained but are not shown in the strain maps as they
hamper the visibility of strains in ferrite.
4. Discussion
Figs. 2(d) and 2(e)show that some regions of ferrite near the phase
boundary have higher KAM values, indicating higher deformation.
These regions are highlighted by dashed white ellipses in Fig. 2(e).
As per the literature, the spatial extent of ferrite deformation lies
approximately in the range of 0.1
l
m–0.5
l
m from the ferrite/-
martensite interface [36]. It is important to note that this distance
is not a single fixed value. Previous micromechanical modeling stud-
ies assuming isotropic dilatation during transformation conclude
that a continuous deformed ferrite region of specific width exists
along the phase boundaries [6,7,18]. The results of the present mod-
eling study, shown in Fig. 6(a) and 6(b), corroborate previous exper-
imental results that the deformed ferrite region is discontinuous and
the extent of deformation varies from one location to another [36].
The PAG reconstruction results shown in Fig. 3 reveal that the
PAGs in the current DP steel specimen are very small with a mean
diameter of 1.6
l
m and an average of 6 variants within every PAG.
As mentioned earlier, several previous studies assumed that the
shear component of transformation for differently oriented vari-
ants cancel each other leaving only the dilatation part as the effec-
tive transformation deformation [6,7,11,12]. However, present
results suggest that such an assumption should be avoided wher-
ever small PAGs with much less than 24 variants are present. For
instance, the prior austenite grain marked ‘PAG 1’ in Fig. 3(a) con-
sist of 14 variants. The magnitude of ferrite deformation adjacent
to ‘PAG 1’ for the case of isotropic dilatation (Fig. 6(a)) is less than
that of anisotropic deformation (Fig. 6(b)). This proves that the
shear deformations accompanying the formation of 14 variants
do not cancel out, otherwise, the transformation-induced deforma-
tion of ferrite would have been similar in both cases. Therefore, the
average transformation deformation in DP steels with small PAGS
cannot be considered equivalent to an isotropic dilatation of prior
austenite.
The micromechanical model calculation results in Figs. 6(a) and
6(b) show that the transformation-induced deformation of ferrite
is non-homogenous throughout the microstructure, regardless of
the assumption of isotropic or anisotropic transformation defor-
mation. This non-homogeneity is more pronounced in Fig. 6(b)
since the variation in magnitude of transformation-induced strains
is larger in this case. For example, the strain in the region marked
‘1’ is a lot higher in magnitude compared to that of the region
marked ‘2’.
The assumption of isotropic or anisotropic transformation
deformation in the micromechanical calculations affects the distri-
bution of transformation-induced strains in ferrite. Large strains
are present at different locations depending on the choice of the
assumption. For instance, in Fig. 6(b), the area marked ‘3’ shows
relatively large strain compared to its surroundings, whereas in
Fig. 6(a) the same area shows comparatively smaller strain. Simi-
larly, the areas marked ‘4’ and ‘5’ also show relatively large strain
in Fig. 6(b) than in Fig. 6(a). In general, at most locations, the strain
in ferrite due to anisotropic transformation deformation is larger
by a factor of up to 10 times than that induced by isotropic
dilatation.
In the case of isotropic transformation deformation, the dilata-
tion is the same for all the variants. Therefore the non-
homogeneity in the transformation-induced deformation of ferrite,
in this case, is because of complex grain shapes. In the case of ani-
sotropic transformation deformation, the non-homogeneity in the
Fig. 6. von-Mises equivalent map of (a) transformation-induced strain in ferrite due to isotropic transformation eigenstrains,

iso;tr-ind
vM
, and (b) transformation-induced strain
in ferrite due to anisotropic transformation eigenstrains,

aniso;tr-ind
vM
.
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
8

transformation-induced deformation of ferrite is not only because
of complex grain shapes but also because of the variants formed
within PAGs. This is because the magnitude and direction of shear
and dilatation deformation are different for every martensitic vari-
ant. As a consequence, every variant uniquely deforms the sur-
rounding ferrite matrix.
On comparing the results of the present micromechanical
model calculations to the GND density map, it is observed that
the regions of high GND density marked ‘II’, ‘III’, and ‘IV’ in Fig. 2
(f) correspond to the regions of high strain ‘4’, ‘6’, and ‘respectively
in Figs. 6(a) and (b). This correspondence is stronger in Fig. 6(b),
i.e., in the case of anisotropic transformation. There are also some
regions of high GND density, such as the one marked ‘I’ in Fig. 2(f),
which do not have high strain at the corresponding locations in the
strain maps. This is perhaps because the transformation-induced
deformation of ferrite observed on the surface is also contributed
to by the martensite formation below the observed surface, while
the present study is carried out on a 2D EBSD-based RVE. Region
‘3’ with high strain in Fig. 6(b) does not have a corresponding
region of high GND density in Fig. 2(f).
This study elucidates the importance of including the aniso-
tropy of martensitic transformation deformation in modeling the
mechanical behavior of DP steels. However, it also reveals that
indirect measures of in-grain deformation in the form of experi-
mental KAM or GND values are insufficient to validate
transformation-induced deformation of ferrite estimated via cur-
rent modeling approach. Hence, the following aspects of this study
can be considered in future investigations: (i) Instead of PTMC,
more advanced theories such as the double shear theory [47] and
parameter-free double shear theory [48] can be used to calculate
anisotropic transformation strains with even more accuracy, (ii)
the local variations in the carbon content of the martensite should
be taken into account while using PTMC and PAG reconstruction,
(iii) 3D PAG reconstruction based on 3D EBSD measurement should
be carried out to take into account the anisotropic transformation
strains associated with martensite formation beneath the observed
surface, (iv) The use of finite strain definition in micromechanical
calculations should provide more accurate quantification of
transformation-induced strains in ferrite.
5. Conclusions
The martensitic transformation-induced deformation of ferrite
has a profound effect on the mechanical behavior of DP steels. In
previous studies modeling the transformation-induced deforma-
tion of ferrite, the anisotropy of transformation deformation was
ignored. Rather, isotropic dilatation alone was considered to
induce the deformation of surrounding ferrite matrix. In this work,
the validity of the assumption of isotropic dilatation during trans-
formation of small PAGs and the effect of anisotropic transforma-
tion deformation in causing deformation of surrounding ferrite
matrix in DP steels is evaluated.
To this end, a methodology is developed whereby the marten-
sitic transformation-induced strains can be studied in DP steels
as well as other multiphase steel microstructures containing
martensite. The methodology includes four steps: (i) Analysis of
the ferrite/martensite dual-phase microstructure, (ii) prior austen-
ite grain reconstruction (iii) determination of martensitic transfor-
mation deformation, and (iv) calculation of transformation-
induced strains in the ferrite matrix surrounding the martensite.
In previous modeling works concerning the transformation-
induced deformation of ferrite in DP steels, PAG orientations were
disregarded. In this study, the PAG orientations and morphology
are obtained via PAG reconstruction from martensite regions of
DP steel microstructure. The results of the PAG reconstruction
and PTMC are then combined with the EBSD orientation data for
the martensite in a unique way to estimate variant-specific trans-
formation deformation required for performing micromechanical
calculations. The calculated transformation-induced strains in fer-
rite are further compared with experimental EBSD KAM and GND
measurements of the same.
The results show regions of relatively large deformation in fer-
rite very close to the phase boundaries in experimental EBSD KAM
and GND density maps as well as simulated ferrite strain maps.
However, not all regions of large deformation in experimental
and simulated results necessarily coincide. The assumption of iso-
tropic or anisotropic transformation deformation in the microme-
chanical calculations affects the distribution of transformation-
induced strains in ferrite. The transformation-induced deformation
of ferrite is determined primarily by prior austenite and ferrite
grain shapes along with the anisotropy of martensitic transforma-
tion deformation. It is also shown that neglecting the shear defor-
mation associated with martensitic transformation under the
assumption of isotropic dilatation results in significant underesti-
mation of transformation-induced strains in ferrite.
CRediT authorship contribution statement
Vibhor Atreya: Conceptualization, Methodology, Software, Val-
idation, Formal analysis, Investigation, Writing – original draft,
Visualization. Jan Steven Van Dokkum: Methodology, Software,
Formal analysis, Writing – review & editing. Cornelis Bos: Writing
– review & editing, Supervision, Project administration. Maria J.
Santofimia: Writing – review & editing, Supervision, Project
administration.
Data availability
The raw and processed data required to reproduce these find-
ings are available to download from: https://doi.org/10.4121/
19411379.
Declaration of Competing Interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgements
This research was carried out under project number T17019j in
the framework of the Research Program of the Materials innova-
tion institute (M2i) (www.m2i.nl) supported by the Dutch
government.
Appendix A
A.1. The micromechanical model
The macroscopic behavior of a homogenized elastic composite
is given as:
S¼C:E;ðA:1Þ
where Cis the effective elastic moduli, Sand Eare the averages of
stress field, rxðÞ, and strain field, xðÞ, respectively, which are com-
puted as the solution of following problem:
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
9

r
xðÞ¼C
r
:

xðÞ;

xðÞ¼

u
þ
xðÞðÞþE;8x2V
r

r
xðÞ¼0;8x2V
u
þ
is periodic ;
r
:^
nis anti-periodic ;
8
>
>
>
<
>
>
>
:
ðA:2Þ
where C
r
is the elastic moduli tensor in phase r;u
þ
xðÞðÞis the
polarization field of strain and V is the periodic volume. To solve
set of Eqs. A.2, a fast fourier transform (FFT) based method is devel-
oped by H. Moulinec and P. Suquet [24] is used. The Problem A.2 is
then replaced by an auxiliary problem which includes a linear
homogenous elastic body of stiffness C
0
under a polarization field
sxðÞ.
r
xðÞ¼C
0
:

xðÞþ
s
xðÞ;
s
xðÞ¼ C
r
C
0

:

xðÞ;

xðÞ¼E
C
0

s
xðÞ;8x2V
r

r
xðÞ¼0;8x2V
u
þ
is periodic;
r
:^
nis anti-periodic ;
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
ðA:3Þ
with the periodic Green’s operator C
0
associated with C
0
and * the
convolution product. The C
0
operator is explicitly known in fourier
space. In the current work, xðÞis initialized with values of transfor-
mation eigenstrains for martensitic variants 

x
ðÞ
. The isotropic
transformation eigenstrain, 
;iso
is given by ðd=3Þ
ii
, where dis
given by Eq. 11, while anisotropic transformation eigenstrains are
calculated from deformation gradient matrix Zobtained via PTMC
theory using the definition of Green-Lagrange strain:

;aniso
xðÞ¼1
2ZxðÞ
T
ZxðÞI
hi
;ðA:4Þ
where I is the identity matrix. Subsequently, 

xðÞis used to calcu-
late the polarization field sxðÞwhich is held constant. Problem A.3 is
then solved iteratively. The convergence is reached when error eis
below a specified value:
e¼k
r

r
k
2
DE
1
2
k
r
hik ;ðA:5Þ
The elastic-plastic von-Mises materials obey the following linear
hardening law:
r
¼C
r



p

;
_

p
¼
3
2
_
p
r
d
r
vM
where p¼
2
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p


p
p;
r
vM
P
r
Y
;
8
>
<
>
:ðA:6Þ
where 
p
is the deviatoric plastic strain, r
d
and
r
vM
are the devia-
toric stress and von-Mises equivalent stress respectively, pis the
hardening parameter and
r
Y
is the yield strength.
References
[1] T. Matsuno, R. Ando, N. Yamashita, H. Yokota, K. Goto, I. Watanabe, Analysis of
preliminary local hardening close to the ferrite–martensite interface in dual-
phase steel by a combination of finite element simulation and nanoindentation
test, Int. J. Mech. Sci. 180 (March) (2020).
[2] F. Badkoobeh, H. Mostaan, M. Rafiei, H.R. Bakhsheshi-Rad, F. Berto,
Microstructural Characteristics and Strengthening Mechanisms of Ferritic–
Martensitic Dual-Phase Steels: A Review, Metals 12 (1) (2022).
[3] S. Li, C. Guo, L. Hao, Y. Kang, Y. An, Microstructure-Based Modeling of
Mechanical Properties and Deformation Behavior of DP600 Dual Phase Steel,
Steel Res. Int. 90 (12) (2019) 1–10.
[4] J.M. Moyer, G.S. Ansell, The Volume Expansion Accompanying the Martensite
Transformation in Iron-Carbon Alloys, Metall. Trans. A 6A (September) (1975)
1785–1791.
[5] J.S. Bowles, J.K. Mackenzie, The crystallography of martensite transformations
I, Acta Metall. 2 (1) (1954) 129–137.
[6] A. Ramazani, K. Mukherjee, U. Prahl, W. Bleck, Transformation-induced,
geometrically necessary, dislocation-based flow curve modeling of dual-
phase steels: Effect of grain size, Metall. Mater. Trans. A 43 (10) (2012)
3850–3869.
[7] A. Ramazani, K. Mukherjee, A. Schwedt, P. Goravanchi, U. Prahl, W. Bleck,
Quantification of the effect of transformation-induced geometrically necessary
dislocations on the flow-curve modelling of dual-phase steels, Int. J. Plast. 43
(2013) 128–152.
[8] U. Liedl, S. Traint, E.A. Werner, An unexpected feature of the stress-strain
diagram of dual-phase steel, Comput. Mater. Sci. 25 (1–2) (2002) 122–128.
[9] D.A. Korzekwa, D.K. Matlock, G. Krauss, Dislocation substructure as a function
of strain in a dual-phase steel, Metall. Trans. A 15 (6) (1984) 1221–1228.
[10] S. Li, C. Guo, L. Hao, Y. Kang, Y. An, In-situ EBSD study of deformation behaviour
of 600 MPa grade dual phase steel during uniaxial tensile tests, Mater. Sci.
Eng., A 759 (March) (2019) 624–632.
[11] T. Sakaki, K. Sugimoto, T. Fukuzato, Role of Internal Stress for the Initial
Yielding of Dual-Phase Steels, Acta Metall. 31 (10) (1983) 1737–1746.
[12] D.L. Bourell, A. Rizk, Influence of martensite transformation strain on the
ductility of dual-phase steels, Acta Metall. 31 (4) (1983) 609–617.
[13] C.C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann, C. Zheng, N.
Peranio, D. Ponge, M. Koyama, K. Tsuzaki, D. Raabe, An Overview of Dual-Phase
Steels: Advances in Microstructure-Oriented Processing and
Micromechanically Guided Design, Annu. Rev. Mater. Res. 45 (2015) 391–431.
[14] R.G. Davies, Early Stages of Yielding and Strain Aging of a, Metall. Trans. A 10A
(1979).
[15] M. Calcagnotto, D. Ponge, E. Demir, D. Raabe, Orientation gradients and
geometrically necessary dislocations in ultrafine grained dual-phase steels
studied by 2D and 3D EBSD, Mater. Sci. Eng., A 527 (10–11) (2010) 2738–2746.
[16] R. Ando, T. Matsuno, T. Matsuda, N. Yamashita, H. Yokota, K. Goto, I. Watanabe,
Analysis of nano-hardness distribution near the ferritemartensite interface in
a dual phase steel with factorization of its scattering behavior, ISIJ Int. 61 (1)
(2021) 473–480.
[17] H. Ashrafi, M. Shamanian, R. Emadi, M. Sanayei, F. Farhadi, J.A. Szpunar,
Characterization of Microstructure and Microtexture in a Cold-Rolled and
Intercritically Annealed Dual-Phase Steel, J. Mater. Eng. Perform. 30 (10)
(2021) 7306–7313.
[18] J. Kadkhodapour, S. Schmauder, D. Raabe, S. Ziaei-Rad, U. Weber, M.
Calcagnotto, Experimental and numerical study on geometrically necessary
dislocations and non-homogeneous mechanical properties of the ferrite phase
in dual phase steels, Acta Mater. 59 (11) (2011) 4387–4394.
[19] C. Du, J.P. Hoefnagels, S. Kölling, M.G. Geers, J. Sietsma, R. Petrov, V. Bliznuk, P.
M. Koenraad, D. Schryvers, B. Amin-Ahmadi, Martensite crystallography and
chemistry in dual phase and fully martensitic steels, Mater. Characteriz. 139
(October 2017) (2018) 411–420.
[20] S. Takaki, K. Fukunaga, J. Syarif, T. Tsuchiyama, Effect of Grain Refinement on
Thermal Stability of Metastable Austenitic Steel, Mater. Trans. 45 (7) (2004).
[21] C. Celada-Casero, J. Sietsma, M.J. Santofimia, The role of the austenite grain size
in the martensitic transformation in low carbon steels, Mater. Des. 167 (2019).
[22] J.S. Bowles, J.K. Mackenzie, The crystallography of martensite transformations
III. Face-centred cubic to body-centred tetragonal transformations, Acta
Metall. 2 (2) (1954) 224–234.
[23] C.M. Wayman, The phenomenological theory of martensite crystallography:
Interrelationships, Metall. Mater. Trans. A 25 (9) (1994) 1787–1795.
[24] H. Moulinec, P. Suquet, A numerical method for computing the overall
response of nonlinear composites with complex microstructure, Comput.
Methods Appl. Mech. Eng. 157 (1–2) (1998) 69–94.
[25] L. Ryde, Application of EBSD to analysis of microstructures in commercial
steels Application of EBSD to analysis of microstructures in commercial steels,
Mater. Sci. Technol. 836 (2006).
[26] G. Miyamoto, N. Takayama, T. Furuhara, Accurate measurement of the
orientation relationship of lath martensite and bainite by electron
backscatter diffraction analysis, Scripta Mater. 60 (12) (2009) 1113–1116.
[27] T. Nyyssönen, M. Isakov, P. Peura, V.T. Kuokkala, Iterative Determination of the
Orientation Relationship Between Austenite and Martensite from a Large
Amount of Grain Pair Misorientations, Metall. Mater. Trans. A: Phys. Metall.
Mater. Sci. 47 (6) (2016) 2587–2590.
[28] T. Nyyssönen, P. Peura, V.T. Kuokkala, Crystallography, Morphology, and
Martensite Transformation of Prior Austenite in Intercritically Annealed High-
Aluminum Steel, Metall. Mater. Trans. A 49 (12) (2018) 6426–6441.
[29] H. Bhadeshia, Worked Examples in the Geometry of Crystals, second ed., The
institute of Metals, London, 1991.
[30] P.M. Kelly, J. Nutting, The martensite transformation in carbon steels, Proc.
Roy. Soc. Lond. Ser. A. Math. Phys. Sci. 259 (1296) (1960) 45–58.
[31] Z. Nishiyama, Martensitic Transformation, Academic Press, Inc., 1972.
[32] X.F. Gu, T. Furuhara, W.Z. Zhang, PTCLab: Free and open-source software for
calculating phase transformation crystallography, J. Appl. Crystallogr. 49
(April) (2016) 1099–1106.
[33] H. Kitahara, R. Ueji, M. Ueda, N. Tsuji, Y. Minamino, Crystallographic analysis of
plate martensite in Fe-28.5 at.% Ni by FE-SEM/EBSD, Mater. Charact. 54 (4–5)
(2005) 378–386.
[34] H. Kitahara, R. Ueji, N. Tsuji, Y. Minamino, Crystallographic features of lath
martensite in low-carbon steel, Acta Mater. 54 (5) (2006) 1279–1288.
[35] L. Sharma, R.H. Peerlings, M.G. Geers, F. Roters, Microstructural influences on
fracture at prior austenite grain boundaries in dual-phase steels, Materials 12
(22) (2019) 1–22.
[36] V. Atreya, C. Bos, M.J. Santofimia, Understanding ferrite deformation caused by
austenite to martensite transformation in dual phase steels, Scripta Mater. 202
(2021) 114032.
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
10

[37] V.A. Lobodyuk, Y.Y. Meshkov, E.V. Pereloma, On Tetragonality of the
Martensite Crystal Lattice in Steels, Metall. Mater. Trans. A 50 (1) (2019) 97–
103.
[38] M. Kamaya, Assessment of local deformation using EBSD: Quantification of
accuracy of measurement and definition of local gradient, Ultramicroscopy
111 (8) (2011) 1189–1199.
[39] F. Bachmann, R. Hielscher, H. Schaeben, Texture analysis with MTEX- Free and
open source software toolbox, Solid State Phenom. 160 (2010) 63–68.
[40] R. Hielscher, T. Nyyssönen, F. Niessen, A.A. Gazder, The variant graph approach
to improved parent grain reconstruction, Materialia 22 (March) (2022)
101399.
[41] F. Niessen, T. Nyyssönen, A.A. Gazder, R. Hielscher, Parent grain reconstruction
from partially or fully transformed microstructures in MTEX, J. Appl.
Crystallogr. 55 (2022) 180–194.
[42] W. Pantleon, Resolving the geometrically necessary dislocation content by
conventional electron backscattering diffraction, Scripta Mater. 58 (11) (2008)
994–997.
[43] A.H. Pham, T. Ohba, S. Morito, T. Hayashi, Effect of Chemical Composition on
Average c/a’ Orientation Relationship in Carbon and Low Alloy Steels, Mater.
Today: Proc. 2 (2015) S663–S666.
[44] A. Saha Podder, H.K. Bhadeshia, Thermal stability of austenite retained in
bainitic steels, Mater. Sci. Eng., A 527 (7–8) (2010) 2121–2128.
[45] R.-M. Rodriguez, I. Gutiérrez, Unified Formulation to Predict the Tensile Curves
of Steels with Different Microstructures, Mater. Sci. Forum 426–432 (2003)
4525–4530.
[46] S.A. Kim, W.L. Johnson, Elastic constants and internal friction of martensitic
steel, ferritic-pearlitic steel, and a-iron, Mater. Sci. Eng. A 452–453 (2007)
633–639.
[47] P.M. Kelly, Crystallography of lath martensite in steels, Mater. Trans. JIM 33 (3)
(1992) 235–242.
[48] F. Maresca, W.A. Curtin, The austenite/lath martensite interface in steels:
Structure, athermal motion, and in-situ transformation strain revealed by
simulation and theory, Acta Mater. 134 (2017) 302–323.
V. Atreya, J.S.V. Dokkum, C. Bos et al. Materials & Design 219 (2022) 110805
11