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Acta Materialia 61 (2013) 2828–2839
www.elsevier.com/locate/actamat
Memory effects of transformation textures in steel and its prediction
by the double Kurdjumov–Sachs relation
T. Tomida a, M. Wakita a,⇑, M. Yasuyama a, S. Sugaya b, Y. Tomota b, S.C. Vogel c
a
Steel Research Laboratories, Technical Research & Development Bureau, Nippon Steel & Sumitomo Metal Corporation, 1-8 Fuso-cho, Amagasaki,
Hyogo 660-0891, Japan
b
Institute of Applied Beam Science, Graduate School of Science and Engineering, Ibaraki University, 4-12-1 Nakanarusawa-machi, Hitachi, Ibaraki
326-8511, Japan
c
Los Alamos Neutron Science Center, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Received 12 August 2012; received in revised form 8 January 2013; accepted 8 January 2013
Available online 13 March 2013
Abstract
The phenomenon that the transformation texture near the initial texture reproduces after the phase transformation cycle such as ferrite
(a, body-centered cubic) ! austenite (c, face-centered cubic) ! a is called a texture memory. In this study, the texture change in a 0.1% C–
1% Mn hot-rolled steel sheet during the a ! c ! a transformation cycle was studied via neutron diffraction and the transformation texture prediction based on a variant selection rule that we call the double Kurdjumov–Sachs (K–S) relation. The texture change observed by
neutron diffraction, which clearly showed the texture memory, could be quantitatively reproduced by the proposed variant selection rule
adopted into the calculation method based on the spherical harmonics expansion of orientation distribution functions. Therefore, it is
most likely that the texture memory in steel is caused by the preferential selection of those K–S variants that reduce the interfacial energy
between a precipitate and two adjoining parent phase grains at the same time, which we call the double K–S relation.
Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Low carbon steel; Texture memory effect; Phase transformation; Orientation relationship; Neutron diffraction
1. Introduction
Crystallographic texture of materials often remains
rather unchanged after the phase transformation cycle
from one phase to another and back to the initial phase
by temperature variation, a phenomenon that is referred
to as texture memory [1,2]. Such a phenomenon has been
reported for various materials such as iron and steel [2–
10], titanium [11], zirconium [12,13] and even minerals like
quartz [1,14], in which the crystallographic orientation
relationship is held between parent and product phases.
This implies that from several to dozens of crystallographically equivalent orientations, which are allowed under the
orientation relationship, only specific ones occur after the
⇑ Corresponding author. Tel.: +81 6 6489 5721.
E-mail address: [email protected] (M. Wakita).
phase transformation, a corresponding phenomenon called
variant selection. Without such variant selection, one
would expect the texture to blur during such phase transformation cycles. However, it has been confirmed microscopically that the martensite (a0 ) in steel can transform
into c, recovering the original orientation of the prior c
during the a0 ! c reverse transformation [3]. It is also
known in diffusive transformation that the texture is often
well maintained through phase transformation cycles, such
as a ! c ! a in steel [4–10] and hexagonal close-packed ! body-centered cubic ! hexagonal-close packed in
Ti [11] and Zr [12,13], whereas grain structures greatly alter
during those cycles. Nevertheless, the mechanism causing
this memory effect is still unclear. Although the influence
of the variant selection concerning the orientation relationship has been inferred as this mechanism, the selection rule
has not been clarified enough. The purpose of this study is
1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.actamat.2013.01.015
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
to clarify the mechanism of the texture memory during the
a ! c ! a transformation cycle in steel by combining
in situ high temperature bulk texture measurements by
neutron diffraction and the texture prediction based on
the proposed variant selection rule [15–18] and spherical
harmonic expansion (SHE) of orientation distribution
functions (ODF) [19].
The texture memory during the a ! c ! a transformation was first studied via ODF analysis in detail by Inagaki
and Kodama [4,5] for hot-rolled 0.08% C–2.2% Mn–Ni–
Mo–Nb steel. It was reported that the texture with strong
{3 1 1}h0 1 1i and {3 3 2}h1 1 3i orientation components was
retained after heating to 910 °C and cooling to the ambient
temperature either by water quench or air cool with an
unexpectedly small decrease in intensity. This phenomenon
clearly indicated that the variants permitted by the orientation relationship between a and c occurred unequally to stabilize the texture through the two phase transformations.
Note that the orientation relation between a and c is known
to be close to the Kurdjumov–Sachs (K–S) relation [20],
{1 1 1}c//{1 1 0}a and h1 1 0ic//h1 1 1ia, or the Nishiyama–
Wasserman (N–W) relation [21,22], {1 1 1}c//{1 1 0}a and
h1 1 0ic//h1 0 0ia, and the K–S and N–W relations have as
many as 24 and 12 crystallographic variants, respectively.
Yoshinaga and co-workers [6,7] later reported similar memory effects for cold-rolled steel as well, in which almost perfect memory effect was observed in Mn-added interstitialfree (IF) steel. The transformation texture and the texture
memory in steel have been reviewed by Ray and Jonas
[23] and Hutchinson et al. [2], respectively.
Several possible mechanisms have been proposed such as
the ones related to the transformation strain [6], the precipitates to stabilize a at an elevated temperature [7] and the special boundaries formed upon heating [2,8]. They all involve
some microstructural sites by which the variant that has
acted on the first transformation is actually “memorized”
and the same path becomes more available than the others
for the backward transformation on cooling. Therefore,
according to these mechanisms, the variant selection for
the texture memory should occur only on cooling, although
the variant selection that can act on both processes should
not be excluded from the possible mechanisms, as has been
inferred in the study of the texture memory in Ti [11] and
Zr [13]. In fact, Bru¨ckner and Gottstein [10] and Wenk and
2829
co-workers [9] have recently conducted the high temperature
diffraction measurements for textures in cold-rolled steel
sheets via X-ray and pulsed neutrons respectively, and they
have reported that variant selection has intensely acted not
only on cooling but also on heating. This fact indicates that
the texture memory might not be caused by the memory of
the variant path during the first transformation but rather
be caused by the two more-or-less “independent” variant
selections in two successive phase transformations.
On the variant selection for the c M a transformation in
steel, the present authors have proposed a selection rule in
which the variants that hold the K–S relation or the near
K–S relation with two adjoining parent phase grains preferentially nucleate (see Fig. 1) [15–18]; we call this selection
rule the double K–S relation hereafter. For the double K–S
relation to be able to be satisfied at the majority of ordinary boundaries, the relation on one side of the grain
boundary is allowed to deviate up to 10° from the exact
K–S relation. By this selection rule, the texture of the
hot-rolled sheet steel has been quantitatively reproduced
by the calculation from the texture of retained c in the
same steel. Furthermore, quite recently, during the preparation of this paper, Lischewski and Gottstein [24] have
reported by an in situ observation of orientation relation
via electron back scattering diffraction (EBSD) that the
double K–S relation is indeed held between a and c in steel
as hypothesized in the literature [15–18], and they could
reproduce the texture memory by this relation. However,
as the EBSD measurement always does, their texture measurement has suffered from insufficient statistics due to a
small number of observed grains, and it might have been
influenced by the presence of surfaces in whose vicinity
the product phase nucleated and grew. Therefore, this
study aimed to thoroughly investigate the phenomenon
of texture memory in steel by neutron diffraction and the
analysis based on the double K–S relation.
2. Experiment procedures
The hot-rolled steel sheet with the chemical composition
listed in Table 1 was prepared using a hot-rolling simulator
[16]. The hot-rolling was finished at about 820 °C, which
was just above the calculated paraequilibrium Ae3 for the
steel (818 °C). Immediately after hot-rolling, the steel sheet
Fig. 1. Schematic representations of (a) the double K–S relation and (b) its influence on variant selection.
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T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
Table 1
Chemical composition of the steel used (mass%).
C
Si
Mn
P
S
0.10
0.05
1.11
0.018 0.003
3. Transformation texture calculation adopting the double
K–S relation
Al
N
0.017
0.0035
Fig. 2. Schematic representation of the temperature change during the
neutron diffraction experiment.
was rapidly cooled to about 650 °C to suppress the recovery of deformed structure, so that the transformation texture from the deformed c could evolve in a. In order to
reduce the influence of the through-thickness variation of
texture, which was caused by the shear deformation due
to friction between sample and roll surfaces, the hot-rolled
steel sheet of 1.2 mm in thickness was thinned to 0.3 mm by
the chemical polishing for the central layer to be remained.
About 30 sheets of 10 10 0.3 mm3 in dimension cut
from the thinned sheet were laser-welded together to create
a cubic stack of 10 10 10 mm3.
Using the cubic stack, the texture change during the
a ! c ! a transformation cycle was investigated via the
neutron diffractometer HIPPO [25] at the Los Alamos
Neutron Science Center. The texture measurements were
first performed at room temperature, then twice at 875 °C
after full a ! c transformation and finally at 400 °C after
cooling, as illustrated in Fig. 2. Four rotations (0°, 45°,
67.5° and 90°) around the vertical axis were measured to
accumulate 108 neutron diffraction histograms per temperature. The time period needed for each texture measurement was about 30 min. The diffraction data were
analyzed by the Rietveld method with the program package MAUD [26]. The ODF was then calculated by the separate SHE-based program [27] based on the recalculated
pole figure data exported from MAUD. The SHE in the
ODF calculation was truncated at the 22nd order. The
microstructures of the specimens before and after the diffraction measurements were examined by etching polished
cross-sectional surfaces with nital and scanning electron
microscopy (SEM). To investigate the microstructure of c
at 875 °C, the as-hot-rolled steel sheet was separately
heat-treated at 875 °C for 30 min, and it was quenched into
iced water to cause martensite transformation. The microstructure of c prior to the martensitic transformation was
then examined by etching polished cross-sectional surfaces
with picric acid solution and SEM.
The crystallographic orientation of parent c and that of
transformed a in low-carbon low-alloy steel in general satisfy the K–S orientation relationship. This relationship is
represented as a rotation of 90° about the h1 1 2i axis of c
crystals as shown in Fig. 3. Therefore, if only one variant
of the possible 24 variants in the K–S relation is activated,
the texture change due to the transformation is a simple
rotation of the texture of the parent phase. Based on this
notion as well as the SHE-based representation of ODF,
Sargent [28] and Davies et al. [29,30] showed that the texture change due to such transformation in which all of
the variants equally operated could be expressed by simple
linear algebraic equations.
Although it becomes rather complicated as many variants are activated with different probabilities, the texture
change due to such transformation can be also dealt by
the calculation method based on the SHE of ODF. In the
SHE-based method [31], the texture transformation due
to the c ! a transformation is expressed as follows:
a
C lm
k ¼
"
ðk1 Þ
1 Mðk
X
X1 ÞNX
c
1
C lm
k1
k1 ¼0 l¼1 m1¼1
ðk2 Þ X
1 N
k X
k
X
X
: m1 l
: ls #
2
qrm
k2 Ak1 ðk1 k2 mrjksÞfk1 k2 m1 m2 jkmgT k ðDgÞ
k2 ¼0 m2 ¼1 m¼ks¼k
s ¼ m þ r; jk2 k1 j 6 k 6 jk2 k1 j; s 6 k
a
C lm
k
c
ð2Þ
C lm
k
and
are the series expansion coefficients of
where
the ODFs of a and c respectively. The Dg represents the
above-mentioned crystal rotation due to the K–S relation,
2
and qr;m
coefficients
of the variant seleck2 ’s are the expansion
:
:
m1 l
tion function q(g). T ls
ðgÞ’s
and
A
’s
are
the symmetrically
k1
k
invariant functions and related coefficients defined for orthorhombic sample symmetry, and (k1k2mr|ks) and {k1k2m1m2|km}
are the Clebsh–Gordan coefficient and the generalized one
defined by Bunge, respectively [19]. Note that although a distribution in Dg is dealt with in Ref. [31], such a distribution is
ignored in the above equations. The a ! c reverse transformation can also be described in the same fashion.
The q(g) in this study is derived from the double K–S
relation [15–18]. Let us now consider how the double
Fig. 3. Schematic representation of the crystal rotation due to the K–S
relation.
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
K–S relation influences the texture development during the
c ! a transformation. We first consider the statistical
probability qi(g) (i = 1, 2, . . . , 24) in which a c grain of orientation “g” transforms into the ith K–S variant of a as
shown in Fig. 1a. This a particle nucleates on a grain
boundary and grows into c1. Then, to satisfy the double
K–S relation, the a particle has to hold the K–S relation
to c2 on the opposite side as well. The more the probability
to find such K–S related c2, the larger the possibility qi(g).
Hence qi(g) should be in proportion to the probability to
find such K–S related c2 as follows:
X
qi ðgÞ /
f ðDg1
k Dg i gÞ
ð3Þ
k
where f(g) is the orientation distribution function of c, Dgi
represents the crystal rotation due to the ith K–S variant of
the c ! a transformation and therefore Dg1
k Dgi g represents the allowed orientations of c2 under the double K–S
relation. As shown in Fig. 3b, if the allowed orientations of
c2 coincide with major texture components in c, qi(g) becomes large; the ith variant is preferentially activated.
The variant selection is thus greatly influenced by the texture of c. To maintain the material balance during the texture transformation calculation in the SHE-based method,
qi(g) should fulfill the equation
24
1 X
q ðgÞ ¼ 1
24 i¼1 i
ð4Þ
Therefore this probability function is scaled for the random selection of variants to be unity.
For the SHE-based method, we rewrite Eq. (3) as
xX
qðgÞ ¼
f ðDg1 gck Dg gÞ þ qC ðgÞ
ð5Þ
N k
x XX
f ðDg1 gck Dg gci gÞ
ð6Þ
qC ðgÞ ¼ 1 24 N i k
where gci ’s are the 24 rotational operators for the cubic
symmetry group [19], and Dg represents one of Dgi’s.
Now the 24 different qi(g) functions are described by one
function q(g). This is possible, since the 24 different variants can be represented by 24 crystallographically equivalent parent orientations, {gci g}, which coexist in the
entire Euler angle space. Although the subscripts, k and
i, for gci and gck can vary from 1 to 24 for the K–S relation,
only 21 variants are taken into account for gck in this study.
The variant in which gck is equal to unity is omitted, since
the orientation of c2 has the equivalent orientation to that
of c1; i.e., the grain boundary for a to precipitate on
disappears. The other two cases in which c2 has a close orientation to c1 (10.5° of misorientation) are also omitted,
since these cases correspond to precipitation on small angle
boundaries. Thus, N in Eqs. (5) and (6), which is the number of k’s, is 21 in this study. It should be noted that the
only free parameter x in Eqs. (5) and (6) determines what
fraction of the material obeys the selection rule due to the
double K–S relation. As x is zero, the selection rule is obvi-
2831
ously unactivated. On the other hand, as x becomes close
to unity, the selection rule becomes fully activated. Therefore q(g) is well defined in the range of x from zero to
unity, whereas it is ill-defined in the range of x over unity,
and q(g) may have physically undesired negative values.
The meaning of x will be further discussed below (see Section 5). Eq. (1) was truncated at the order of 22nd, while
the expansion of q(g) was truncated at the order of 16th,
which ensured the tolerance for the K–S relation at 10°
(see Fig. 1).
For the a ! c transformation, the q(g) is defined likewise considering the double K–S relation in which a and
c are interchanged in Fig. 1a. The texture transformation
during the phase transformation cycle of a ! c ! a has
been thus calculated using Eqs. (1), (2), (5), and (6) as well
as their equivalents for the reverse transformation. In addition, the reverse untransformation calculation, in which the
texture of the parent phase is deduced from the texture of
the product phase and which is often called “untransformation”, has been performed to investigate the prior c (cpre) to
the initial a, i.e., the c just after hot-rolling during sample
preparation. The untransformation calculation was made
by solving Eq. (1) with Eq. (5), which is quadratic and
needs a convergent method to solve [18].
4. Results
The observed texture change during the a ! c ! a
transformation cycle measured via HIPPO is presented in
Fig. 4 as a change of the ODF sections at 0° and 45° in
/2 of the Euler angle. Since the two consecutively measured
textures of c at 875 °C are almost exactly the same, only the
first one is shown in the figure. By comparing the initial and
final textures, the texture memory in which the a texture of
the as-hot-rolled steel is almost completely recovered
through the two phase transformations is confirmed. The
main components of the a texture lie in the regions extending from {1 0 0}h0 1 1i to {2 1 1}h0 1 1i and from {2 1 1}h0 1 1i
to {3 3 2}h1 1 3i, which are known to develop in the texture
transformed from deformed c in hot-rolled steel except for
{1 0 0}h0 1 1i [23]. On the other hand, the main components
of the texture in the c at 875 °C lie around {1 1 0}h1 1 2i and
{2 1 1}h1 1 1i, which are typical of the rolling (deformation)
texture in face centered cubic (fcc) metals with medium to
high stacking fault energy including c [23,32]. Therefore,
the texture of c at 875 °C resembles the texture of c expected
for the sample resulting from hot-rolling during sample
preparation, and therefore it is likely that the texture memory persists not only in the a ! c ! a transformation cycle
but also in the prior cpre ! a ! c transformation cycle, in
which cpre is the deformed c just after hot-rolling. It is also
worth noting that since little difference has been observed
between the two consecutively measured textures of c, the
texture of c has been fairly stable at 875 °C. This agreement
also lends credibility to the reproducibility of our experimental results.
2832
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
Fig. 4. The texture change in hot-rolled 0.1% C–1% Mn steel measured with the neutron diffractometer HIPPO depicted in ODF sections of 0 and 45° in /2.
Fig. 5. ODF sections of textures of c (a) measured with HIPPO and (b) calculated based on the double K–S relation from the experimental texture of
initial a.
The mechanism of the texture memory was then investigated based on the comparison between the experimental
texture by HIPPO and the calculated one by the SHE-based
method in which the double K–S relation was adopted. The
ODF sections of the texture of c measured at 875 °C by
HIPPO and those calculated from the initial a texture measured at the room temperature are shown in Fig. 5a and b,
respectively. The main components of both textures are
seen to correspond well in peak positions and intensity in
all sections. The comparison in fiber plots in the area from
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
2833
Fig. 6. ODF sections of textures of a after a ! c ! a transformation cycle: (a) measured with HIPPO and (b) calculated based on the double K–S relation
from the experimental texture of c.
Fig. 7. Comparison of fiber plots between calculated and experimental textures: (a) a ! c with rising temperature and (b) c ! a with falling temperature.
{1 1 0}h0 1 1i to {1 1 0}h0 0 1i are also shown in Fig. 7a.
Whereas some discrepancy can be found such as slightly
smaller peak intensity in the calculated texture, the overall
features of textures after the first a ! c transformation
are quantitatively well reproduced by the calculation.
In Fig. 6 are shown the experimental texture of a after
the full transformation cycle and the corresponding texture
of a reproduced by using the measured textures of c shown
in Fig. 5a. Comparing the two textures, the accordance
between the measured and calculated ones is again obvious
in all areas of ODF. Slightly better agreement than that in
the first transformation is observed in the skeleton lines in
the area from {1 0 0}h0 1 1i to {1 1 0}h0 1 1i (see Fig. 7b).
Therefore, it is clear that the texture memory can be quantitatively predicted by the SHE-based analysis in which the
double K–S relation is introduced as a variant selection
rule.
Additional fiber plots for the textures calculated without
selection rules are shown in Fig. 7, for which the parameter
x in Eq. (5) is nil. This gives us the idea about how the introduction of the double K–S relation improves the prediction.
Although it has greatly improved the prediction of the
2834
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
Fig. 8. Deviation, ðDf 2 Þ1=2 , of the predicted texture from the experimental one as a function of the parameter x: (a) a ! c with rising temperature and (b)
c ! a with falling temperature.
Fig. 9. Probabilities of activation of variants for the calculation shown in Fig. 5b, a ! c with rising temperature.
Fig. 10. Probabilities of activation of variants for the calculation shown in Fig. 6b, c ! a with falling temperature.
texture change on cooling, the improvement is not so prominent on heating. This is simply because the value of the
parameter x needed for the best prediction is only 0.45 on
heating, whereas that on cooling is as large as 1.4. Note
that, as shown in Fig. 8, the values of x have been determined in such a way that the average of squared deviation
between the measured and calculated textures is minimized.
Hence the calculation suggests that although the variant
selection has certainly operated in both directions of transformation, the variant selection on heating has been weaker
than that on cooling, which is in good agreement with the
results reported in the previous studies [9,10].
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
2835
Table 2
Examples of probabilities of variant selection in the calculations shown in Figs. 5b and 6b, rotation axes of variants and approximated orientations of
product phases. Major orientation components in product phases are shown with bold and italic letters.
Variant
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Rotation axis
Transformation from a to c
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
Transformation from c to a
{3 3 2}h1 1 3ia
{3 1 1}h0 1 1ia
{1 1 0}h1 1 2ic
{2 1 1}h1 1 1ic
Orientation in
c
Probability
Orientation in
c
Probability
Orientation in
a
Probability
Orientation in
a
Probability
{3 2 1}h1 1 1i
{3 1 1}h1 1 2i
{3 1 1}h1 1 2i
{2 1 1}h1 1 1i
{3 1 1}h1 1 2i
{3 2 1}h1 1 1i
{3 2 2}h2 2 1i
{3 2 1}h1 1 1i
{3 1 1}h1 1 2i
{3 2 2}h2 2 1i
{3 2 1}h1 1 1i
{3 2 1}h1 1 1i
{3 2 1}h1 1 1i
{3 1 1}h1 1 2i
{3 1 1}h1 1 2i
{2 1 1}h1 1 1i
{3 1 0}h0 0 1i
{3 2 0}h0 0 1i
{3 1 0}h0 0 1i
{3 2 0}h0 0 1i
{3 1 0}h0 0 1i
{3 1 0}h0 0 1i
{3 2 0}h0 0 1i
{3 2 0}h0 0 1i
1.108
1.077
1.004
1.163
1.030
1.057
0.918
1.131
1.030
0.918
1.057
1.131
1.108
1.004
1.077
1.163
0.791
0.881
0.945
0.897
0.791
0.945
0.881
0.897
{1 1 0}h1 1 2i
{3 2 1}h1 0 3i
{3 2 1}h1 2 1i
{1 1 0}h1 1 2i
{2 2 1}h1 0 2i
{1 1 0}h1 1 2i
{4 4 1}h1 2 4i
{2 1 0}h1 2 1i
{2 2 1}h1 0 2i
{4 4 1}h1 2 4i
{1 1 0}h1 1 2i
{2 1 0}h1 2 1i
{1 1 0}h1 1 2i
{3 2 1}h1 2 1i
{3 2 1}h1 0 3i
{1 1 0}h1 1 2i
{2 1 0}h1 2 0i
{3 1 0}h1 3 0i
{2 1 0}h1 2 0i
{3 1 0}h1 3 0i
{2 1 0}h1 2 0i
{2 1 0}h1 2 0i
{3 1 0}h1 3 0i
{3 1 0}h1 3 0i
1.346
1.034
1.215
1.036
0.791
1.358
0.938
1.163
0.791
0.938
1.358
1.163
1.346
1.215
1.034
1.036
0.779
0.891
0.658
0.791
0.779
0.658
0.891
0.791
{1 1 1}h1 2 3i
{2 1 1}h1 1 3i
{3 2 2}h1 1 3i
{2 1 1}h1 2 0i
{2 1 1}h1 1 3i
{1 1 1}h1 2 3i
{2 1 1}h1 1 3i
{1 1 1}h1 1 2i
{1 0 0}h0 1 1i
{1 0 0}h0 1 1i
{1 0 0}h0 1 2i
{1 0 0}h0 1 2i
{1 1 0}h0 1 2i
{1 0 0}h0 1 1i
{1 0 0}h0 1 1i
{1 0 0}h0 1 2i
{2 1 1}h1 1 3i
{1 1 1}h1 2 3i
{2 1 1}h1 1 3i
{1 1 1}h1 1 2i
{3 3 2}h1 1 3i
{2 1 1}h1 1 3i
{1 1 1}h1 2 3i
{2 1 1}h1 2 0i
0.497
1.294
1.201
0.759
0.888
0.853
1.019
0.964
1.506
1.455
0.812
0.752
0.812
1.455
1.506
0.752
1.019
0.853
0.888
0.964
1.201
1.294
0.497
0.759
{2 1 1}h0 1 1i
{4 2 1}h0 1 2i
{4 2 1}h0 1 2i
{3 2 1}h1 3 3i
{4 2 1}h0 1 2i
{3 1 1}h0 1 1i
{3 2 1}h0 1 2i
{3 1 1}h0 1 1i
{4 2 1}h0 1 2i
{3 2 1}h0 1 2i
{3 1 1}h0 1 1i
{3 1 1}h0 1 1i
{2 1 1}h0 1 1i
{4 2 1}h0 1 2i
{4 2 1}h0 1 2i
{3 2 1}h1 3 3i
{2 1 0}h1 2 0i
{1 1 0}h1 1 0i
{2 1 0}h1 20 i
{1 1 0}h1 1 0i
{2 1 0}h1 2 0i
{2 1 0}h1 2 0i
{1 1 0}h1 1 0i
{1 1 0}h1 1 0i
1.630
1.194
1.463
0.795
0.362
1.941
0.631
1.546
0.362
0.631
1.941
1.546
1.630
1.463
1.194
0.795
0.538
0.876
0.264
0.760
0.538
0.264
0.876
0.760
The probabilities of variant selection in those calculations are exemplified in Figs. 9 and 10 and Table 2, in
which the probabilities are normalized as their average
for all of the 24 variants becomes unity. The correspondence between the variant numbers and the h1 1 2i rotation
axes of the K–S relation as well as the approximated product orientations after transformation are listed in Table 2.
It is recognized that almost all of the variants have been
activated in every case. In the calculation for the transformation on heating with the value of x being 0.45, the probability varied only in the narrow range from 0.8 to 1.3 for
the parent orientations of {3 1 1}h0 1 1ia and {3 3 2}h1 1 3ia
as seen in Fig. 9, and the global maximum and minimum
for all of the parent orientations were 1.51 and 0.57 respectively. In the calculation for the transformation on cooling
with the larger value of x being 1.4, the probability varied
in the relatively wide range from 0.25 to 2 for the parent
orientations of {1 1 0}h1 1 2ic and {2 1 1}h1 1 1ic as shown
in Fig. 10, and the global maximum and minimum for all
of the parent orientations probabilities were 2.48 and
0.08 respectively; although the value of x was over unity,
there was barely negative probability because of the relatively weak parent texture. Thus, even in the transformation on cooling, more than a half of 24 variants were
assigned to work with a probability of more than half of
random probability in the calculation. It is also observed
in Table 2 that the probabilities for the variants that
transform into the major orientations of product phases
such as {2 1 1}h1 1 1ic and {1 1 0}h1 1 2ic for the a ! c transformation, and {3 1 1}h0 1 1ia and {2 1 1}h0 1 1ia for the
c ! a transformation always hold larger values than unity.
Hence, the texture memory can be explained by two variant
selection events, in which variant selection is statistically
determined only by the texture of the parent phase, and a
relatively large number of variants simultaneously operate.
The texture of cpre deduced from the observed initial
texture is shown in Fig. 11. The value of x for this untransformation calculation was chosen to be 0.7, since the previous study suggested that the proper value of x for the
c ! a transformation after hot-rolling should be from 0.6
to unity [15–18]. The deduced texture has the developed
b-fiber [23,32] with large density at the {1 1 0}h1 1 2i and
{2 1 1}h1 1 1i orientations, which resembles that in the
observed texture at 875 °C and typical of the rolling texture
of c. Therefore the expected texture memory in the cpre ! a ! c transformation cycle is reproduced by the present
untransformation calculation as well.
Fig. 12 displays the microstructures of the sample before
and after the heat cycle of the neutron diffraction measurement as well as that of the sample separately heat-treated at
875 °C. Note that the grain structure in the intermediate
figure, which has been revealed by etching martensitic
structure with picric solution, displays the structure of
prior c at 875 °C. It is seen that, although the grain size
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T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
Fig. 11. ODF sections of textures of cpre deduced based on the double K–
S relation from the experimental texture of initial a.
of initial a is small at about 4 lm, those of c at 875 °C and
final a are as large as about 20 lm. It means that rather
substantial grain growth has occurred on heating, perhaps
mostly after the full transformation into c, whereas it is not
the case on cooling. The possible alteration of texture during this grain growth, which is not taken into account on
the present calculation, may, at least partly, account for
the deviation of the calculated texture from the experimental one on heating, (see Figs. 5 and 7a).
5. Discussion
Conventionally the texture memory has been ascribed to
the residual stresses due to transformation strain on heating that bias transformation on cooling to return to the original orientation, the dissolution of Ti or Nb from carbide
and nitride precipitates in IF steel that makes a part of a
remain and grow on cooling, and the special boundaries
formed on heating that are suited for the a of original orientation to nucleate. All those mechanisms involve some
microstructural mechanisms that promote transformation
back to the original orientation as mentioned earlier. Contrarily to this, it has been shown in this study that the phenomenon called texture memory can be quantitatively
predicted by the purely statistical mechanism based on
the orientation relationship between product phase and
adjoining parent phase grains, which does not involve a
direct mechanism to memorize the original orientation at
all. It reproduces the texture change on heating, which cannot be accounted for by the above conventional models, as
well as the texture change on cooling remarkably well.
Moreover, the aspect that the texture memory becomes
more prominent as the initial texture becomes severer [8] is
also naturally ascribed to the present model due to the double K–S relation, since the variant selection depicted by
Eqs. (3) and (5) strengthens as the initial texture sharpens.
One way to scrutinize the validity of the present model is
to investigate if it persists in the texture memory starting
with a very different initial texture. For this purpose, the
texture memory in cold-rolled ultra-low carbon steel, in
which a strong {1 1 1} texture reappears after a ! c ! a
transformation, was employed, and its texture change
was examined based on the present model. The used data
were those measured via HIPPO by Wenk and co-workers
[9] in a way similar to that in the present experiment. The
pole figures provided by those authors, which were
exported from MAUD [26] and Beartex [33], were reanalyzed by the SHE-based software [27] to obtain ODFs,
and the same calculation as shown above was performed.
The results are presented in Figs. 13 and 14. It is clearly
seen that the simulation has successfully reproduced the
texture memory in the cold-rolled ultra-low carbon steel
quantitatively. The degree of reproduction of texture in this
simulation is even better than that in the hot-rolled steel
presented above. It is also very interesting to note that
the components around {1 0 0}h0 3 2i seen in these figures,
which are scarcely present in the initial texture, but emerge
in the final texture, and thus do not pertain to the texture
memory, are also reproduced by the simulation well. Those
results firmly corroborate the present variant selection
model.
Fig. 12. Microstructures (left and right) before and after neutron diffraction measurements and (intermediate) after separate heat treatment at 875 °C. The
averaged grain diameters are shown as da and dc.
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
2837
Fig. 13. Measured texture changes in cold-rolled ultra-low carbon steel during a ! c ! a transformation cycle after Ref. [8]: (a) the texture of initial a at
800 °C, (b) the texture of intermediate c at 950 °C, (c) the texture of final a at 400 °C.
Fig. 14. Calculated textures based on the double K–S relation from the experimental textures of cold-rolled ultra-low carbon steel: (a) positions of major
orientations, (b) the texture of c calculated from Fig. 12a, and (c) the texture of a calculated from Fig. 12b.
The values of x for this simulation, which were determined in the same fashion shown in Fig. 8, were 0.9 and
1.8 on heating and cooling respectively. Although the values of x are larger than those for the hot-rolled steel, the
tendency that the required parameter value on cooling is
larger than that on heating is the same for both cases;
i.e., the variant selection is stronger on cooling than on
heating. The value of x is basically structure-dependent.
In the following, the meaning of these parameter values
is discussed.
The value of x, which determines the intensity of variant
selection as depicted by Eq. (5), can be regarded as a fraction
of parent phase that obeys the double K–S relation on transformation. Even though the double K–S relation is favored
on transformation, the relation will not be kept if there is
nowhere to be able to nucleate having the double K–S relation. The probability to find such a nucleation site suitable to
satisfy the double K–S relation is about 70% on ordinary
boundaries, if the maximum tolerance angle for the near
K–S relation on one side of the boundary is assumed to be
10° [17]. This means that at 30% of grain boundaries, it is
impossible for the product phase to nucleate with the double
K–S relation. Then the transformation should take place
ignoring the double K–S relation, or the nucleation never
occurs on such boundaries. Therefore we expect that the
value of x is from 0.7 (the former case) to unity (the latter
case). In fact, the proper values of x from 0.6 to unity have
been reported for the c ! a transformation after hot-rolling
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T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
[15–18]. The values of x on heating, 0.45 for hot-rolled steel
and 0.9 for cold-rolled steel are thus reasonable values, with
taking into account that the observed grain growth on heating for the hot-rolled steel (see Fig. 12) may modify the
apparent intensity of variant selection by altering the texture. In other words, the value of x on heating might be
somewhat underestimated due to the texture alteration associated with the grain growth.
On the other hand, the value of x over unity on cooling
is beyond the boundary where the present model becomes
ill-defined. It might be attributed to the nucleation on triple
junctions of grain boundaries in the parent phase, which
the present model does not explicitly take into account.
At the triple junction, the nucleation of the product phase
may occur having the K–S relation with three adjoining
parent gains at the same time to further reduce interphase
boundary energy. In such a case, the Eq. (3) may be rewritten as
X
2
qi ðgÞ /
ðf ðDg1
k Dgi gÞÞ
ð7Þ
k
which may act as if the value of x in the Eq. (5) is increased
to over unity. Certainly, Lischewski and Gottstein reported
for the transformation on heating that the nucleation of
product phases mostly occurred at the triple junctions,
and the product phase particles often held the near K–S
relation with all the neighboring parent grains [24]. On
cooling, the transformation temperature is always somewhat decreased than that on heating due to the delay of
transformation. The lower the transformation temperature,
the more the influence of interphase boundary energy on
variant selection becomes [8]. Therefore, on cooling, the
influence of the nucleation at triple junctions on the variant
selection should become more prominent than that on
heating, which even more increases the value of x apparently. This type of nucleation would be able to occur at
15% of the triple junctions in an ordinary grain structure,
if the deviation angle from the exact K–S relation is allowed up to 10°, while it has been reported that 10.3%
of nucleation has been observed to be of this type for the
a ! c transformation in steel [24].
Finally, it is worth discussing the influence of special
boundaries and the difference between its related mechanism [2,8] and the present one. The special boundary model
for the texture memory is that as two different K–S variants
of c emerge from one parent grain on heating, the boundary between the c grains is the one at which an a grain of
the original orientation can nucleate on cooling having the
exact K–S relation to both of neighboring c grains, as
shown in Fig. 15. Hence such special boundaries operate
as an orientation memory site on cooling. In this model,
the conventional K–S relation with a narrow tolerance
(usually 2–3° from the exact K–S relation) is implicitly
assumed, and no influence of the nucleation on the general
boundaries is taken into account.
The above model may be regarded as a special case of
the present model. However, a significant difference to be
Fig. 15. Schematic representation of one of the conventional mechanisms
of texture memory due to special boundaries, (upper) two nucleating c
particles having the K–S relation to the grain a1 on an a grain boundary
on heating, (intermediate) a c grain structure with a special boundary after
completion of a ! c transformation, and (bottom) a nucleating a grain
with the original orientation of a1 having the double K–S relation on the
special c boundary on cooling. V1 and V2 represent two variants of the K–
S relation. The special boundaries would disappear by the growth of c
grains.
noted is that the relatively large tolerance from the K–S
relation assumed in the present model permits many different variants from the one operated on heating to be operative on cooling. It also allows the nucleation to occur on
general boundaries with keeping the double K–S relation,
and accordingly it lets the texture of the parent phase statistically determine the variant selection as described in
Eqs. (5) and (6). On the other hand, the special boundary
model with little tolerance relies entirely on the density of
such boundaries to determine the variant selection instead
of the texture of the parent phase. It then requires a different way to predict textures than Eqs. (5) and (6) and leaves
the origin of the variant selection on heating unanswered as
mentioned above.
Nevertheless, if this special boundary frequently exists,
the probability for the double K–S relation to return to
the original orientation to be fulfilled certainly increases,
then the value of x may apparently rises. Therefore, this
model can be a factor for the larger parameter value on
cooling. When the c grains grow far over the grain size
of initial a, the special boundary, however, should disappear, since only one of the emerged c grains probably survives. This is probably the case in this experiment, since the
grain size of c is about five times larger than that of initial a
as shown in Fig. 12. Therefore, although it might be operating simultaneously with the present model, the influence
of this mechanism is likely to be limited.
6. Conclusions
The texture memory during the a ! c ! a transformation cycle of the hot-rolled 0.1% C–1% Mn steel sheet was
T. Tomida et al. / Acta Materialia 61 (2013) 2828–2839
investigated via neutron diffraction, and the mechanism of
the texture memory was studied by the SHE-based analysis
in which the double K–S relation was introduced as a variant selection rule. The results are summarized as follows:
(1) The texture memory in the hot-rolled steel sheet was
successfully confirmed by the neutron diffraction
measurement. The texture that had the major components lying around {1 0 0}h0 1 1i, {3 1 1}h0 1 1i and
{3 3 2}h1 1 3i reappeared after the transformation
cycle with little decrease in intensity.
(2) The high temperature texture measurement at 875 °C
revealed that the texture of c after the a ! c transformation had the characteristics of fcc rolling textures.
It was then suggested that the texture memory persisted not only the a ! c ! a transformation cycle
but also the c ! a transformation during the prior
hot-rolling process to prepare the sample.
(3) The observed texture memory was quantitatively predicted by the SHE-based analysis in which the double
K–S relation was adopted as a variant selection rule.
Thus the interfacial energy should play a major role
in the phenomenon of texture memory, and the phenomenon is likely to be caused not by the presence of
sites memorizing active variants in the transformation on heating but by two independent variant selections due to the double K–S relation. The texture of
parent phase statistically determines the variant selection at each phase transformation leading to the texture memory.
(4) The proposed mechanism was assessed by examining
the texture memory in the cold-rolled steel with a different type of texture as well, and the calculation with
the mechanism successfully reproduced it, including
the aspect in the observed textures that cannot be
ascribed to the memory of active variants on heating.
(5) The variant selection is stronger on cooling than
heating, although on heating the estimated strength
of variant selection by calculation might be influenced by the possible change of texture due to grain
growth. Especially the strength of variant selection
on cooling exceeds the limit predicted with the double
K–S relation on nucleation at general boundaries,
and it indicates the rather frequent nucleation at triple junctions.
Acknowledgements
The authors gratefully acknowledge that Prof. H.-R.
Wenk in University of California, Berkeley and his coworkers have kindly allowed the use of the data by HIPPO
on the texture memory in cold-rolled steel. They also thank
Dr. N. Imai, Mr. Y. Tanaka and Mr. M. Yoshida of Steel
2839
Research Laboratory of Nippon Steel & Sumitomo Metal
Corporation for valuable discussion.
This work has benefited from the use of the Lujan Neutron Scattering Center at LANSCE, which is funded by the
Office of Basic Energy Sciences, U.S. Department of Energy. Los Alamos National Laboratory is operated by
Los Alamos National Security LLC under DOE Contract
No. DE-AC52-06NA25396.
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