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5 FiguresSequential {1012} twinning stimulated by other twins in titanium
Abstract
We experimentally studied sequential twinning in rolled pure titanium at room temperature and identified a new sequential twinning mechanism, i.e., ( ) extension twin stimulated by the twintwin junction of two primary {11 } extension twins and , which holds a certain relation → . Schmid factor analysis, as the local stress is taken into account, is able to determine the position of sequential twin variant, but cannot determine the twin variant. Displacement gradient accommodation is used to determine the twin variant. The wellknown secondary {10 2} ( ) extension twins in primary {11 2} ( ) contraction twins (referred to as → double twinning) are also analyzed according to a generalized Schmid factor analysis. Displacement gradient accommodation and twin nucleation based on dislocations dissociation only work well for the most active twin variants, but cannot address other phenomena associated with → double twinning. For rarely activated twin variants, displacement gradient accommodation was not satisfied.
Figures
Full length article
Sequential f1012gtwinning stimulated by other twins in titanium
Shun Xu
a
,
b
,
c
, Mingyu Gong
c
, Christophe Schuman
a
,
b
,
**
, JeanS
ebastien Lecomte
a
,
b
,
Xinyan Xie
c
, Jian Wang
c
,
*
a
Laboratoire d'Etude des Microstructures et de M
ecanique des Mat
eriaux (LEM3), CNRS UMR 7239, Universit
e de Lorraine, F57045, Metz, France
b
Laboratory of Excellence on Design of Alloy Metals for lowmAss Structures (DAMAS), Universit
e de Lorraine, France
c
Mechanical and Materials Engineering, University of NebraskaLincoln, Lincoln, NE 68588, USA
article info
Article history:
Received 29 December 2016
Received in revised form
7 April 2017
Accepted 9 April 2017
Available online 18 April 2017
Keywords:
Titanium
Twinning
Variant selection
Twintwin junction
abstract
We experimentally studied sequential f1012gtwinning in rolled pure titanium at room temperature and
identiﬁed a new sequential twinning mechanism, i.e., f1012g(TI
i) extension twin stimulated by the twin
twin junction of two primary {1121} extension twins TII
i1and TII
i, which holds a certain relation
TII
i1TII
i/TI
i. Schmid factor analysis, as the local stress is taken into account, is able to determine the
position of sequential twin variant, but cannot determine the twin variant. Displacement gradient ac
commodation is used to determine the twin variant. The wellknown secondary {1012} (TI
j) extension
twins in primary {1122} (CI
i) contraction twins (referred to as CI
i/TI
jdouble twinning) are also analyzed
according to a generalized Schmid factor analysis. Displacement gradient accommodation and twin
nucleation based on dislocations dissociation only work well for the most active twin variants, but
cannot address other phenomena associated with CI
i/TI
jdouble twinning. For rarely activated twin
variants, displacement gradient accommodation was not satisﬁed.
©2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Titanium with hexagonal close packed structure (referred to as
a
Ti) is used in different industries [1e4].
a
Ti plastically deforms
via slips and twins. A tremendous amount of experimental work
has been carried out for
a
Ti and other hexagonal metals to un
derstand mechanisms and mechanics of slips and twins in the
context of temperatures and strain rates [5,6], cyclic loading [7e9],
strain path changes [10], textures [11,12], twinning modes [13],
grain size effects [14,15], and sample size effects [16]. In addition to
slips associated with basal <a>slips f0002g1120, prismatic <a>
slips f1100g1120, pyramidal <cþa>slips f1011g1123, and pyra
midal <a>slips f1101g1120 at room temperature [17,18], twinning
is a prevalent deformation mechanism to accommodate imposed
strains. f1012g1011 extension twinning and f1122g1123 contrac
tion twinning were most commonly observed at room temperature
[19,20]. Other twinning modes, f1121g1126 extension twinning
[21e24],f1124g2243 contraction twinning, and f1011g1012
contraction twining [25,26], occur depending on temperature and
loading condition. Extension twinning relates to the aspect of
introducing a positive strain along the caxis of the parent grain and
contraction refers to a negative component along the caxis.
Twinning in hexagonal materials is directional with a unique sense
of shear and occurs at low shear stress [27]. A localized shear
deformation associated with twinning results in mechanical
instability of hexagonal metals [28,29]. As a consequence of the
polarity of twinning, rolled hexagonal metal plates exhibit a char
acteristic texture [30]; the ﬂow stress evolution shows a strong
anisotropy between the inplane and throughthickness directions
[31,32]; and aggregated twinning shows an increase in hardening
rate and a continuous evolution of grain microstructure with
deformation [33,34]. In particular, during cyclic loading or strain
path changes, a twin interacts with another twin, resulting in the
formation of twintwin junctions that inﬂuence subsequent plastic
deformation modes, slips, twinning, secondary twinning, and de
twinning [35e38]. Thus, there is also an urgent demand for the
development of predictive capabilities that can describe twinning
and twinninginduced sequential events, and their correlations
with microstructures, temperatures, and loading conditions [5,6].
These predictive models will enable engineers to optimize me
chanical forming processes of hexagonal metals for speciﬁc
*Corresponding author.
** Corresponding author. Laboratoire d'Etude des Microstructures et de
M
ecanique des Mat
eriaux (LEM3), CNRS UMR 7239,Universit
e de Lorraine, F57045,
Metz, France.
Email addresses: christophe.schuman@univlorraine.fr (C. Schuman),
jianwang@unl.edu (J. Wang).
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
http://dx.doi.org/10.1016/j.actamat.2017.04.023
13596454/©2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Acta Materialia 132 (2017) 57e68
applications [39e42].
The core issues in developing physicsbased predictive model
include where a twin is initiated, how a certain twin variant is
selected, and when twinning occurs [43e47]. According to the
sequence of nucleating twin variants in a grain during mechanical
loading, deformation twins can be classiﬁed into primary twins,
secondary twins, and tertiary twins. Corresponding to the statisti
cal nature of twin nucleation, nucleation occurs when the local
resolved shear stress on a given twin variant exceeds the statisti
cally assigned nucleation threshold stress [48]. Schmid factor (SF)
was extensively used to determine the selection of primary twin
variants with the assumption that the local stress is the same as the
applied one [49]. However, the local stresses indeed differ from the
applied one [50]. This is consistent with the phenomena that
twinning is associated with negative SFs in experiments [51,52].To
consider the inﬂuence of local stresses on twin nucleation, a
random stress that is related to grain orientation, grain size and
grain boundaries, is recently added to the applied one [53e55].
Slightly differing from primary twinning, secondary twinning ex
hibits a strong correlation between the secondary twin variant and
primary twins. Several deterministic criteria for the selection of
twin variants have been proposed for some speciﬁc circumstances,
for example, slip transmission induced twin variant [56e58] and
twin transmission induced twin variant [25,59e61] across grain
boundaries. Correspondingly, the SF criterion was completed with
the displacement gradient accommodation (DGA) [51,62], the
deformation energy [63], and the nucleation of twinning disloca
tions based on dislocation dissociation (NDD) [64]. These criteria
have been demonstrated validate in accounting for secondary
f1012gtwins in {1011} primary twins in Mg [65e68], secondary
f1012gtwins in primary f1012gtwins [69e71],f1012gtwinning
inside {1122} primary twins in
a
Ti and Zr [72e75], and secondary
twinning or retwinning during detwinning as primary twintwin
junction is subjected to reversal loading in Mg [38].
In this paper, we experimentally studied sequential f1012g
twinning in rolled pure titanium at room temperature. In addition
to the wellknown secondary f1012gextension twins in primary
{1122} contraction twins (referred to as {1122}/f1012gdouble
twinning), we found a new sequential {1012} extension twin that
was stimulated by the primary {1121} twintwin junction (referred
to as f1121gTT Junction/f1012g). We examined the correlation
between sequential f1012gtwin and primary twins, and found that
the selection of sequential twin variant can be predicted by the
combination of the three criteria dSchmid factor (SF), displace
ment gradient accommodation (DGA), and the nucleation of
twinning dislocations based on dislocation dissociation (NDD).
These sequential twinning mechanisms could be implemented into
meso and macro scale predictive models [34].
2. Experiments
The rolled commercially pure titanium T40 sheet (ASTM grade
2) with the thickness of 1.5 mm was annealed in a vacuum furnace
at 800
C for 3 h. After annealing, the sheet was fully recrystallized
with an average grain size of ~200
m
m and no twins appear. The
annealed sheet was subjected to a compressive strain of 7% at a
strain rate 1.0E3 s
1
at room temperature using a Zwick 120T
machine. The compression direction is along the ND. After the
compression, the surface of the deformed sample was ground with
SiC papers of grits from 1200
#
to 4000
#
. Electrolytic polishing was
performed using a solution of 10% perchloric acid and 90% meth
anol at 35 V for 5 s at 5
C for EBSD measurements, which were
applied on a JEOL JSM6500F ﬁeld emission gun scanning electron
microscopy (SEM) equipped with an EBSD camera and the AZtec
acquisition software package (Oxford Instruments). During that,
the sample was tilt by an angle of 70
and a voltage of 15 kV was
used. The EBSD patterns for the detailed information on twins were
acquired at a step size of 0.2
m
m. The data that indicates next
neighbor grain to grain misorientation was processed by using
ATOM software [76].
Fig. 1a shows EBSD patterns of the polished surface with a step
size of 0.5
m
m. Fig. 1b shows twin boundaries that are characterized
with a tolerance of ±5
deviation from the ideal crystallographic
axis and angle. Corresponding to the crystallography of twins in
a

Ti, {1121} twinning rotates the twinned domain by ~35
around a
<1100>axis; f1012gtwinning rotates the twinned domain by ~87
around a <1210>axis; {1122} twinning rotates the twinned
domain by ~64
around a <1100>axis; {1124} twinning rotates the
twinned domain by ~77
around a <1100>axis, with respect to the
parent. We statistically analyzed the nextneighbor grain to grain
misorientation (Fig. 1c). The peaks at ~35
and ~64
are attributed
to the formation of {1121} and {112 2} twins, respectively. The
{1124} contraction twins do not produce an obvious peak at ~77
due to their small volume fraction. The maximum peak at ~87
is
associated with f1012gextension twins. f1012gtwins are activated
either as primary twin in grain (G1 in Fig. 1b) or as secondary twin
associated with {1122}/f1012gdouble twinning (G2 in Fig. 1b)
and f1121gTT Junction/{1012} twinning (G3 in Fig. 1b and d).
{1122} contraction twins are either only one twin variant in grains
(G4 in Fig. 1b) or coexistent with secondary f1012gextension twin
(G2 in Fig. 1b). {1121} extension twins in grains are either as single
twin variant (G5 in Fig. 1b), coexistent with f1012gextension twin
(G6 in Fig. 1b) or associated with two variants interaction (G3 in
Fig. 1b). The interesting ﬁnding is that the interaction between two
{1121} extension twin variants stimulates f1012gtwins.
In what follows, we identify secondary f1012gtwins in EBSD
patterns according to the crystallography of twins and then analyze
the selection of secondary f1012gtwin variant according to Schmid
factor (SF), displacement gradient accommodation, twin nucle
ation, and the local stress associated with twintwin interactions.
3. f1121gTT Junction/f1012gtwinning
3.1. Crystallographic character
In hexagonal structure, 12 rotational symmetry matrices result
in 6 equivalent variants for each twinning system. We denote twin
variants in Fig. 2 as T
I
i
for six f1012gtwin variants and T
II
i
for six
{1121} twin variants where i ¼1…6. The subscript iincreases by a
counterclockwise rotation around the caxis of the crystal. The
zone axis 1010 associated with a T
II
i
twin variant is the vector
summation of two zone axes
1
3
1120 associated with T
I
i
and T
I
iþ1
twins. The red and blue arrows indicate the twinning directions of
the two twins.
To characterize the f1012gtwin variant stimulated by {1121} TT
junction, the EBSD patterns in Fig. 3a and b were acquired at a step
size of 0.2
m
m. In order to identify the twin variants, six {1121}
planes in the matrix and in the twin are plotted into the pole ﬁgure
corresponding to the orientation of the observed grain (Fig. 3c). The
misorientation angle between ð1121Þplane in T
II
1
twin and in the
matrix is 1.05
, and the misorientation angle between ð1211Þplane
in T
II
2
twin and the matrix is 2.06
. Thus the two primary twin
variants are (1211Þ½1216(T
II
1
) and (1121Þ½1126(T
II
2
). Dashed lines
in Fig. 3c mark the traces of the two twin planes. The sequential
extension twin variant is determined according to the pole ﬁgures
of {1012} planes and of 0111 twinning directions. The misorien
tation angle between ð0112Þplanes in the twin and in the matrix is
0.23
, and between ð0112Þtwin planes in the twin and in the
matrix is 10.08
, as shown in Fig. 3d. The misorientation angle
between 0111 directions in the T
I
2
twin and the matrix is 1.04
, and
S. Xu et al. / Acta Materialia 132 (2017) 57e6858
that between 0111 directions in the T
I
5
and in matrix is 10.03
as
shown in Fig. 3e. According to the crystallography of twins {1012}
<1011>, twin variant (0112Þ½0111(T
I
2
) is a good ﬁt. The red, green
and black dashed lines in Fig. 3a indicate the traces of the three
twin planes on the observed surface. The three twins hold a relation
of T
II
1
T
II
2
/T
I
2
.
There are several geometry features associated with T
II
1
T
II
2
/T
I
2
sequential twinning. The common axis between T
II
1
and T
II
2
is par
allel to <0113>as indicated by a dashed line in Fig. 3f. A twintwin
boundary (TTB) produced by the f1121gtwintwin junction is
identiﬁed to lie in (2110) plane in the grain (denoted by the while
line in Fig. 3f and g) with the assumption that the interface bisects
two intersected twinning planes of T
II
1
and T
II
2
twins [77]. We con
ducted statistical analysis in our samples. Among 46 {1121} twin
twin junctions, we found that the obvious TTB always forms in
the obtuse corner and lie in {2110} planes. f1012gtwin variant is
also always located at the obtuse region between the two primary
twinning planes. We identiﬁed 96 f1012gtwins associated with 46
{1121} twintwin junctions. They hold the relationship that T
I
i
forms in the obtuse region between T
II
i1
and T
II
i
, where the zone
axis associated with T
I
i
is a shared vector in describing the zone axis
of the two primary twins. Thus such a sequential twinning mech
anism can be described by T
II
i1
T
II
i
/T
I
i
.
To conﬁrm such a sequential twinning mechanism, we con
ducted quasi insitu EBSD analysis by applying two steps of
compression along the ND on a prepolished sample. In the ﬁrst
step, a reduction of 6.2% was applied, followed by EBSD charac
terization with a step size of 0.2
m
m. As shown in Fig. 4a, {1121}
twintwin junctions were generated. Then, second step of
compression was done with a further reduction of 3.3% along the
same direction (ND), followed by EBSD mapping on the same po
sition. Fig. 4b shows that new f1012gtwins were activated in the
obtuse region. In order to identify the twin variants, six {1121}
planes in the matrix and in the twin are plotted into the pole ﬁgure
corresponding to the orientation of the observed grain. The
misorientation angle between (1121) plane in T
II
4
twin and in the
matrix is 1.15
, and the misorientation angle between (1211) plane
in T
II
5
twin and the matrix is 1.06
. Thus, the two primary twin
Fig. 1. (a) EBSD patterns of the deformed Ti, (b) the nextneighbor grain to grain misorientation analysis, (c) the distribution of grain misorientation angles, and (d) the magniﬁed
EBSD pattern of the region G
3
in (b). The yellow and red lines indicate {1121} and f1012gtwin boundaries; the blue and pink lines indicate {1122} and {1124} twin boundaries. (For
interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 2. The crystallography of twins in hexagonal structure. Four shadow planes are
twin planes associated with two {1121} twins, TII
1(light green) and TII
2(yellow), and
two f1012gtwins TI
2(pink) and TI
5(green). Two red arrows indicate the twinning
shear directions of {1121} twins, and the blue arrows represent the twinning shear
directions of f1012gtwins. (For interpretation of the references to colour in this ﬁgure
legend, the reader is referred to the web version of this article.)
S. Xu et al. / Acta Materialia 132 (2017) 57e68 59
variants are (1121)[1126] ( T
II
4
) and (1211)[ 1216] (T
II
5
). Dashed lines
in Fig. 4c mark the traces of the two twin planes. The sequential
f1012gextension twin variant is determined according to the pole
ﬁgures of f1012gplanes in Fig. 4d and of 0111 twinning directions
in Fig. 4e. The misorientation angle between (0112) planes in the
twin and in the matrix is 0.96
, and between (0112) twin planes in
the twin and in the matrix is 6.16
, as shown in Fig. 4d. The
misorientation angle between 0111 directions in the T
I
5
twin and
the matrix is 1.04
, and that between 0111 directions in the T
I
2
and
in matrix is 6.95
as shown in Fig. 4e. According to the crystallog
raphy of {1012}<1011>twins, twin variant (0112Þ½0111(T
I
5
)isa
good ﬁt. The SF of T
II
4
and T
II
5
twins in Fig. 4 is 0.424 and 0.369,
respectively. They carry the highest and second highest SF among
six {1121} variants. The SF of active T
I
5
and absent T
I
4
is 0.277 (the
second highest) and 0.345 (highest), respectively. The three twins
hold a relation of T
II
4
T
II
5
/T
I
5
.
3.2. Selection criteria for T
II
i1
T
II
i
/T
I
i
To understand such a sequential twinning mechanism, we
further conduct the analysis of SF, displacement gradient accom
modation, and local stress ﬁelds, associated with the T
II
i1
and T
II
i
junction.
3.2.1. Location selection of sequential twin
Why the sequential twin always presents in the obtuse region is
ﬁrstly examined according to stress ﬁelds due to the two primary
twins using ﬁnite element method (FEM). In the FEM model shown
in Fig. 5a, two {1121} twin domains T
II
1
and T
II
2
were embedded in a
4040 8 matrix through the thickness along the zaxis in the
model. In order to conveniently analyze the resolved shear stress on
twin planes associated with sequential twinning, the matrix adopts
the coordinates with the xaxis along ½1210, the yaxis normal to
ð3032Þand the zaxis along ½1013. The z direction is parallel to the
intersection line of T
II
1
and T
II
2
twin planes. Correspondingly, the
trace of the twin plane associated with the sequential twin variant
T
I
2
or T
I
5
is parallel to the xaxis or the yaxis. Two twins have an
elliptical shape in the crosssectional plane along the zaxis. For T
II
1
twin, the long axis was 22.7
m
m and short axis was 3.7
m
m. For T
II
2
twin, the long axis was 15.8
m
m and the short axis was 3.1
m
m.
Anisotropic elastic modulus was assigned into the three regions
corresponding to the local orientation [78]. Eigenstrains that mimic
the corresponding twinning shear were applied to two twin do
mains. The model was meshed with 127328 hexahedron elements
for better accuracy. To obtain the feature of the elastic stress ﬁeld,
here we conducted linear elastic calculation with free boundaries
while ignoring plasticity relaxation associated with slips or twins in
the matrix. A thin slice in the middle of the model depicted in
Fig. 5a was chosen for the analysis of backstress that is caused by
the two primary twins. Due to the near 90
between the two twin
planes associated with twin variants T
I
2
and T
I
5
, the resolved shear
stress associated with T
I
2
and T
I
5
twins are approximately same. We
only show the resolved shear stress ﬁeld associated with twin
variant T
I
2
in the matrix in Fig. 5b. Fig. 5c shows the variation of the
resolved shear stress along two directions in the acute and obtuse
region. The result shows that the resolved shear stress associated
with twin variants T
I
2
and T
I
5
are negative, i.e., opposite to the twin
shear direction. However, the resolved shear stress due to the back
stress is smaller in the obtuse region than in the acute region. Under
an applied stress, the net resolved shear stress in the obtuse region
is thus greater than that in the acute region, favouring sequential
extension twinning in the obtuse region. However, the stress
analysis could not identify the twin variant between T
I
2
and T
I
5
.
3.2.2. SF analysis
Corresponding to the crystallography of twins in Fig. 3, SFs
associated with T
I
5
and T
I
2
twins are found to be 0.277 and 0.201,
respectively. T
I
5
and T
I
2
variants carry the highest and second
highest SF, but the second high SF twin T
I
2
was only activated. It is
noticed there is only one speciﬁc loading direction along [0110]
under which T
I
2
and T
I
5
have the largest and identical SF. There is a
question whether the activated twin variant is always associated
with the second high SF among six twin variants. Without loss of
generality, SF analysis is further conducted under uniaxial
compressive stresses that favor two primary twins T
II
i1
and T
II
i
among six twin variants. All loading directions that satisfy the
condition are grouped into one loading domain (referred to as the
T
II
i1
T
II
i
loading domain). Fig. 6a shows the T
II
i1
T
II
i
loading domain in
which the T
II
1
and T
II
2
twins are subjected to the highest and sec
ondary higher SF among six {1121} twin variants. The red means
that both twins have SFs near 0.5, orange and pink mean that the SF
of one twin is close to 0.5. We also checked our samples and found
that the loadings acting on the grains that have two {1121} twin
variants are in the loading domain. The black pentagram in Fig. 6a
indicates the loading direction of the grain G3 along [0.0501
0.4705e0.5206 0.3197]. Secondly, we calculate SFs associated with
six T
I
i
twins as the grain is subjected to uniaxial compression within
Fig. 3. (a) and (b) EBSD characterization of f1012gtwins and {1121} twintwin
junctions as well as the pole ﬁgure of (c) {1121} plane, (d) f1012gplane, (e) 1011
direction with respect to the orientation of the observed grain. In (c), the black dots
represent six {1121} twin planes in the matrix, the blue squares and red stars represent
six {1121} twin variants in two twins. In (d), the black dots represent six {1012} twin
planes in the matrix, the red diamonds represent six f1012gtwin planes in the twin,
and (e) the black dots represent six 1011 twin directions in the matrix, the red di
amonds represent six 1011 twin directions in the twin, (f) and (g) Schematics of
f1121gTT Junction /f1012gsequential twinning. (For interpretation of the refer
ences to colour in this ﬁgure legend, the reader is referred to the web version of this
article.)
S. Xu et al. / Acta Materialia 132 (2017) 57e6860
the loading domain. Due to the approximate 90
between the twin
planes of T
I
i
and T
I
iþ3
, a pair of T
I
i
and T
I
iþ3
is subjected to the same
resolved shear stress. Fig. 6b, c and 6d show the SFs associated with
T
I
1
and T
I
4
,T
I
2
and T
I
5
, and T
I
3
and T
I
6
, respectively. The blue color
represents negative SFs, and red represents positive SFs. The SFs
associated with T
I
2
and T
I
5
are always greater than that for other two
pairs, implying that T
I
2
and T
I
5
are mechanically preferred if
sequential twin occurs. However, the result does not determine the
preference between the two variants. We further compare the SFs
associated with the two twin variants and found that the SF asso
ciated with T
I
5
is always higher than the SF associated with T
I
2
for
any applied compression in the loading domain (Table 1). This is
contradictory to EBSD analysis where T
I
2
always occurs. Therefore,
the SF analysis can determine the group of sequential twins, but
cannot help determine the sequential twin variant.
3.2.3. Deformation gradient accommodation
The same shear strain may be associated with different crystal
rotation that depends on displacement gradients. Corresponding to
the simple shear model associated with twinning, we calculate the
deformation gradient resulted from T
II
1
and T
II
2
twinning. The
displacement gradient tensor of a twin has a simple form in the
twinning frame, i.e., x jj the twinning shear direction (
h
), z jj the
normal of the twin plane (n), and y jj the zone axis (
l
¼n
h
)[51].
The displacement gradient tensor can be written as
e
ij
¼2
4
00s
000
000
3
5(1)
where s¼0.63 for {1121} twinning (c/a ¼1.587) in
a
Ti. In order to
examine whether the strain induced by the {1121} twin can be
accommodated by slips and/or other twins in the grain, the
displacement gradient tensors of the intersected {1121} twins were
transformed into the crystal frame of the grain, i.e., x
0
jj [1010], y
0
jj
[1210], z
0
jj [0001]. e
x
0
z
0
and e
y
0
z
0
represent the accommodation by
Fig. 4. Quasi insitu EBSD maps of {1121} twintwin junctions and sequential f1012gtwins with a reduction of: (a) 6.2%, (b) 9.5% as well as the pole ﬁgure of (c) {1121} plane, (d)
f1012gplane, (e) 1011 direction with respect to the orientation of the observed grain. In (c), the black dots represent six {1121} twin planes in the matrix, the blue squares and red
stars represent six {1121} twin variants in two twins. In (d), the black dots represent six {1012} twin planes in the matrix, the red diamonds represent six {1012} twin planes in the
twin, and (e) the black dots represent six 1011 twin directions in the matrix, the red diamonds represent six 1011 twin directions in the twin. (For interpretation of the references to
colour in this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 5. (a) Schematic of the FEM model. (b) Resolved shear stress ﬁelds associated with twin variants TI
2and TI
5in the matrix. (c) The variation of the resolved shear stress (MPa, in
the vertical axis) along two directions R
o
and R
A
(
m
m, in the horizontal axis). The resolved shear stress is caused by the backstress due to the TT junction.
S. Xu et al. / Acta Materialia 132 (2017) 57e68 61
double and single basal slips, respectively; e
z
0
x
0
and e
z
0
y
0
indicate
the accommodation by twinning; e
x
0
y
0
and e
y
0
x
0
are associated with
the accommodation by double and single prismatic slips, respec
tively [51]. For the two primary twins, the transformed displace
ment gradient tensors are
e
ij
T
II
1
¼0
@0:135 0:078 0:049
0:078 0:045 0:028
0:496 0:287 0:181
1
A(2)
e
ij
T
II
2
¼0
@
00 0
00:181 0:057
00:573 0:181
1
A(3)
e
z
0
x
0
and e
z
0
y
0
have big value, implying that dislocation slips could
not efﬁciently accommodate the displacement gradients, instead,
extension twinning corresponding to the positive e
z
0
z
0
could
accommodate them.
We thus evaluate the accommodation capacity of six {1012}
extension twin variants and other four {1121} extension twin var
iants (Table 1). The displacement gradient tensors of the activated
{1121} twins were transformed into the twinning frames of {1012}
variants and {1121} variants, respectively. As summarized in Table 1,
four {1121} twin variants do not accommodate the displacement
gradient component due to the negative sign. The variant T
I
2
has the
largest e
13
among six {1012} and four {1121} twins, implying that
T
I
2
twin variant can effectively accommodate the displacement
gradient component generated by the two {1121} primary twins. T
I
5
twin with the highest SF does not accommodate the displacement
gradient component because of the negative sign. Other twin var
iants, T
I
1
and T
I
4
, can accommodate the displacement gradients,
while they often have small SFs, and even negative SFs (Table 1).
Thus, the combination of SF and DGA criteria assures the relation of
T
II
i1
T
II
i
/T
I
i
.
4. f1122g/f1012gdouble twinning
4.1. Crystallography of f1122g/f1012gdouble twins
For simplicity in describing twins later, we denote twin variants
in Fig. 7 as T
I
i
for six {1012} twin variants and C
I
i
for six {1122} twin
variants where i ¼1, 2, 3 …6. The subscript i increases by a counter
clockwise rotation around the caxis of the crystal. The zone axis
1010 associated with a C
I
i
twin variant is the vector summation of
two zone axes
1
3
1120 associated with T
I
i
and T
I
iþ1
twins. Comparing
the parent grain orientation to secondary twins inside, there are 36
possible combinations. The misorientation between a secondary
{1012} twin originating from a primary {1122} twin and its parent
grain was determined by rotating the orientation of the parent
grain around the respective normal directions of primary and sec
ondary twinning planes by 180
. Then, the 12 symmetry operations
of the hexagonal lattice were applied and the minimum misori
entation angle and corresponding axis were computed between the
orientation matrix of the parent and the 12 equivalent matrices of
the secondary twin. According to misorientation between the
matrix and the secondary twins, six {1012} secondary twins inside
a primary {1122} twin were categorized into 3 groups: C
I
i
/T
I
iþ3
and
T
I
iþ4
(Group I), C
I
i
/T
I
i
and T
I
iþ1
(Group II), and C
I
i
/T
I
iþ2
and T
I
iþ5
(Group III). In Group I, the angle between two twin planes is 95.9
and the angle between two twin shear directions is 76.6
. In Groups
II and III, the angle between two twin planes is 27.4
and 66.9
,
respectively, and the angle between two twin shear directions is
24.2
and 55.2
, respectively.
The crystallographic feature of C
I
i
/T
I
j
double twins is illustrated
in Fig. 8. The blue plane denotes the primary twin C
I
i
, and the pink
plane represents the secondary twin T
I
j
. The white dashed line in
dicates the intersection, l
ij
, between the primary twin C
I
i
and the
secondary twin T
I
j
. The intersection lines, l
14
and l
15
in Group I are
along ½810 2 3and ½10823, respectively. The intersection lines
l
11
and l
12
in Group II are along ½4223and ½2423, respectively. The
same intersection lines are identiﬁed for l
13
and l
16
along ½4223
and ½2423in Group III, respectively. The intersection lines in
Fig. 6. Inverse pole ﬁgures of the Schmid factor of twinning systems subjected to
compression: (a) the loading domain in which SFs of TII
1and TII
2twins are positive and
are greater than other four equivalent variants. Under this loading domain, the SF
associated with sequential twin variants (b) TI
1and TI
4, (c) TI
2and TI
5, and (d) TI
3and TI
6.
The blue color represents negative SFs, and red represents positive SFs. (For inter
pretation of the references to colour in this ﬁgure legend, the reader is referred to the
web version of this article.)
Table 1
The transformation of displacement gradient tensor of T
II
1
and T
II
2
twins into the twinning frame of six {1012} extension twin variants and other four {1121} extension twin
variants in the same grain (Euler angle: 4
1
¼0.7
,
F
¼120.5
,4
2
¼35
) as well as the SFs under macro loading (Exp. SFs) and the SFs under a speciﬁc loading along [0110]
direction (Spe. SFs).
Twin variants Six f1012gvariants Four f1121gvariants
T
I
1
T
I
2
T
I
3
T
I
4
T
I
5
T
I
6
T
II
3
T
II
4
T
II
5
T
II
6
ð1012Þ1011 ð0112Þ0111 ð1102Þ1101 ð1012Þ1011 ð0112Þ0111 ð1102Þ1101 ð2111Þ2116 ð1121Þ1126 ð1211Þ1216 ð2111Þ2116
Spe. SFs 0.125 0.498 0.125 0.125 0.498 0.125 0 0.215 0.215 0
Exp. SFs 0.029 0.201 0.078 0.015 0.277 0.046 0.103 0.224 0.225 0.041
T
II
1
ð1121Þ1126 0.41 0.41 0.09 0.10 0.10 0.09 0.20 0.42 0.20 0.33
T
II
2
ð1211Þ1216 0.09 0.41 0.41 0.09 0.10 0.10 0.33 0.20 0.42 0.20
The bold font indicates positive and greater SFs.
S. Xu et al. / Acta Materialia 132 (2017) 57e6862
Groups II and III lie in a f1010gprismatic plane, as outlined by the
yellow dashed lines.
Experiment observations show that Group I, Group II and Group
III account for 10.6%, 85.6% and 3.8% of 425 detected
f1122g/f1012gdouble twins [75], respectively. In the previous
work [75], the analysis of f1122g/f1012gdouble twins was
applied according to the experimental data. Both intersection lines
of the primary and secondary twin planes lie in one active prismatic
plane in the primary twin, correspondingly Xu. Et al analyzed the
production of twinning partial dislocations. However, this
prismaticdislocation mediated nucleation mechanism is appli
cable to both Group II and Group III double twin variants, and thus
it cannot explain why Group II is preferred over Group III. In this
work, we extend the experimental ﬁndings to a general case to
evaluate whether the prismaticdislocation mediated nucleation
mechanism can distinguish variant selection in all possibilities.
Besides, the dissociation of prismatic dislocations into twinning
dislocations is applied to both Group II and Group III {1012} double
twins. Even though the geometrical feature also facilitates the
activation of Group III double twins, the dislocation reaction pre
sented in this paper clearly shows that a prismatic dislocation
dissociating into three secondary twining dislocations for Group II
twin is energetically favored while a prismatic dislocation disso
ciating into only one twining dislocation associated with Group III
twin is energetically unfavorable, which clearly explains why
Group III is still difﬁcult to happen. Therefore, only prismatic
dislocation mediated nucleation mechanism can clarify the pref
erence of Group II over Group I and Group III. In addition, with the
help of a generalized SF analysis, the stress domains in which only
one or two Group II double twin variants can be activated were in
agreement with experiments. Therefore, a general and more
rigorous prediction of the selection for f1012gdouble twinning
variants inside {1122} primary twins is presented in this work.
Obviously, Group II twins are predominant among three groups
according to statistical analysis. In addition, most primary twins,
95.5% of double twins, contain only one secondary twin variant.
Only 4.5% of primary twins in Group II contain two secondary twin
variants. In what follows, we tested the variants correlation of
{1122 }/{1012} double twins according to SF, displacement
gradient accommodation, and nucleation of twinning dislocations
based on dislocation dissociation.
4.2. Selection criteria for {1122}/f1012gdouble twins
4.2.1. A generalized SF analysis
Without loss of generality, we conducted a generalized SF
analysis. Firstly, we determine the loading domain in which one
primary contraction twin is activated. {1122} twin variant C
I
1
is
chosen to be the primary twin in this study. The grain is subjected
to uniaxial compression. The loading domain is determined ac
cording to the SF associated with C
I
1
twin. When the SF is greater
than 0.3 and is the greatest among six equivalent variants, the
loading directions are plotted into an inverse pole ﬁgure as shown
in Fig. 9a. Due to the approximate 90
between secondary twin
variants T
I
i
and T
I
iþ3
, they have nearly same SF. At a given loading
direction within the loading domain, we calculated the SFs asso
ciated with six f1012gtwin variants T
I
j
(j ¼1…6), and plotted
them in Fig. 9bed. The results show that twin variants T
I
3
and T
I
6
in
Group III have smaller SFs than twin variants in Group II and Group
I. The SFs in Group II and I are similar and no signiﬁcant difference,
because T
I
1
in Group II has the same SF as T
I
4
in Group I, and T
I
2
in
Group II has the same SF as T
I
5
in Group I. This could address the
lower activity of twin variants in Group III than other two groups,
while this could not distinguish the activity of twin variants in
Group II and I. The same feature is associated with other contrac
tion twin variants. For example, the loading domain associated
with C
I
2
is obtained by counterclockwise rotating the C
I
1
loading
domain 60
around the <0001>. Three typical loading directions
are marked by A, B and C in Fig. 9a. SFs of six secondary twin var
iants at the three points are given in Table 2.T
I
1
and T
I
4
have the
largest SFs when the loading is along the A; T
I
2
and T
I
5
have the
largest SFs when the loading is along the C. Four variants in Group II
and Group I have the largest SFs when the loading is along the B.
We further compared SFs associated with twinvariants in Group
II. When the difference in the SFs associated with two variants in
one group is smaller than 0.05, we treated them equal. Corre
spondingly, we replotted the loading domain in Fig. 9e and f for
Group II associated with the primary contraction twin C
I
5
and C
I
4
,
respectively. In a small loading domain (marked by the red color),
two variants share the equal SF. In the rest region of the loading
domain, one is greater than the other. The same feature is also
observed for Group I. This could address why most primary twins
only contain one secondary twin variant.
4.2.2. Displacement gradient accommodation
The generalized SF analysis so far well predicts the activity of
twin variants in Group II, while the competition of variants be
tween Group II and Group I cannot be accessed. We thus tested
displacement gradient accommodation. Displacement gradient
tensor of a ð1122Þ½1123twin expressed in its twinning frame was
transformed into the twinning frame of six {1012} twins. The
component e
13
is summarized in Table 2. Two variants in Group II
have the largest e
13
, which means that the strain induced by the
primary twin can be effectively accommodated by Group II sec
ondary twins. Twin variants in Group I are also observed in ex
periments, but the displacement gradient tensor is negative. As to
twin variants in Group III, the displacement gradient tensor is
positive but they are rarely observed in experiments. This implies
that displacement gradient accommodation is not the necessity in
accessing the variant, but can predict the relative activity when
multiple variants have positive SF in the case of C
I
i
/T
I
j
double twins
Fig. 7. The crystallography of hexagonal structure. Two shadow planes are twin planes
associated with contraction twin {1122}, CI
1(blue), and extension twin f1012g,T
I
1
(pink). (For interpretation of the references to colour in this ﬁgure legend, the reader is
referred to the web version of this article.)
S. Xu et al. / Acta Materialia 132 (2017) 57e68 63
(j ¼ioriþ1).
4.2.3. Nucleation based on dislocation dissociation
The selection of secondary twin in the double twinning could be
related to twin nucleation at primary twin boundary due to dislo
cation dissociation. In Mg, {1011}{1012} were the most commonly
observed DTs under uniaxial deformation. In this case, primary and
secondary twins have the same axis along aaxis. The nucleation of
secondary twins inside the contraction twin was proposed by
Beyerlein et al. to be aided by the dissociation of basal <a>dislo
cations. In Ti, we examine the dislocation dissociation mechanism
associated with prismatic <a>dislocations.
The intersection line between two secondary twin variants in
Group I and the primary twin ð1122Þare along ½8 10 2 3and
½10 8 2 3, respectively. These intersection lines do not lie in any
usual slip planes. Thus, dislocation dissociation mechanisms would
be difﬁcult unless dislocations can easily climb or cross slip on the
twin plane. However, two secondary twin variants in Group II
intersect ð1122Þtwin plane along ½4223and ½2423, respectively.
They both lie in a f1100gprismatic plane. The same characteristic
can be found for two secondary twins variants in Group III.
Therefore, nucleation of secondary twin variants in Group II and III
could be mediated by the dissociation or reﬂection of prismatic
dislocations.
Taking the similar analysis in Ref. [64], we investigate the
dissociation or reﬂection of prismatic <a>dislocation at the pri
mary twin plane into twinning dislocations associated with {1012}
twins. As shown in Fig. 10a, secondary T
I
2
and T
I
3
variants intersect
the C
I
1
primary twin plane along the ½2423axis, which lies in
(1010) prismatic plane. When ð1010Þ1210 dislocations approach
the primary twin boundary as depicted in Fig. 10b, they could
dissociate into twinning dislocations associated with Group II and
Group III secondary twins. For Group II, a ð1010Þ1210 dis
location, b
a
can be dissociated into three secondary twining
dislocations 3b
ð0112Þ
t
, and a residual one b
1
r
,
b
a
0b
1
r
þ3bð
0112
Þ
t
(4)
according to the Frank's law ðjb
a
j
2
>
b
1
r
2
þ
3b
ð0112Þ
t
2
Þ. However, a
ð1010Þ1210 dislocation can dissociate into only one twining
dislocation associated with Group III twin,
b
a
0b
2
r
þbð
1102
Þ
t
(5)
The dissociation is energetically unfavorable according to the
Frank's law, jb
a
j
2
<
b
2
r
2
þ
b
ð1102Þ
t
2
. Where b
a
¼
1
3
½1210, 0.295 nm
for Ti; The magnitude of Burgers vector of {1012} twinning dislo
cation is equal to
3a
2
c
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3a
2
þc
2
p
, 0.60 nm for Ti (a¼0.295 nm and
c¼0.4683 nm).
If the incoming dislocation b
a
is ﬁrstly dissociated into twinning
dislocation b
ð1122Þ
t
on the primary twin plane, and then the residual
acts as nucleation source, only one secondary twinning dislocation
can be nucleated according to the Frank's law.
b
a
0b
3
r
þbð
1122
Þ
t
þbð
0112
Þ
t
(6)
b
a
0b
4
r
þbð
1122
Þ
t
þbð
1102
Þ
t
(7)
Equation (6) was applied for Group II and was energetically
favored. Equation (7) was applied for Group III and was energeti
cally unfavorable. The magnitude of Burgers vector of {1122}
twinning dislocation is equal to
c
2
2a
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
c
2
þa
2
p
, 0.82 nm for Ti.
The above analysis according to the nucleation of twinning
Fig. 8. Crystallography of f1122g/f1012gdouble twins. The blue plane denotes the primary twin CI
i, and the pink plane represents the secondary twin TI
j. The white dashed line
indicates the intersection, lij, between the primary twin CI
iand the secondary twin TI
j. The yellow dashed lines outline a f1010gprismatic plane. (For interpretation of the references
to colour in this ﬁgure legend, the reader is referred to the web version of this article.)
S. Xu et al. / Acta Materialia 132 (2017) 57e6864
dislocations based on the dissociation of prismatic <a>dislocation
can distinguish the relative activity of three group of twin variants.
Group II is predominant, Group III is secondly activate, and Group I
is most unlikely. Due to the approximately same resolved shear
stress on two twin variants in a group, the nucleation criterion
cannot distinguish the preference of two variants in the same
group.
5. Discussion
5.1. T
II
i
1
T
II
i
/T
I
i
sequential twinning
According to EBSD analysis of sequential twins in rolled pure
titanium at room temperature, we observed a new sequential
twinning mechanism that {1012} extension twin is stimulated by
the primary {1121} twintwin junction. Among 71 cases of f1121g
TT Junction/{1012} twinning, experimental observations reveal
that the relation among three extension twins is T
II
i1
T
II
i
/T
I
i
, and T
I
i
is always formed in the obtuse region between two primary twin
planes. Corresponding to the crystallography of twins, six twin
Fig. 9. Inverse pole ﬁgures of the Schmid factor of twinning systems subjected to
compression: (a) Stress domain where SF of CI
1primary twin variant is positive and is
the largest among the six possibilities. Under this stress domain, the SF of: (b) TI
1and
TI
4secondary twin variants, (c) TI
2and TI
5secondary twin variants, (d) TI
3and TI
6sec
ondary twin variants. Distribution of SFs of Group II variants TI
iand TI
iþ1in the loading
domain associated with contraction twin (e) CI
5and (f) CI
4. Green means the SF asso
ciated with variant TI
iis larger than the variant TI
iþ1. Blue means the SF associated with
variant TI
iis smaller than the variant TI
iþ1. Red means the difference in the SFs of two
variants is less than 0.05. (For interpretation of the references to colour in this ﬁgure
legend, the reader is referred to the web version of this article.)
Table 2
Comparison of selection criteria for {1122}/{1012} double twins. SFs associated with six twin variants correspondto three loading directions A, B and C in Fig. 9a. DGA means
displacement gradient accommodation, and NDD represents nucleation based on dislocation dissociation.
In primary twin (C
I
1
)½11221123
Secondary f1012g
variants
T
I
4
T
I
5
T
I
1
T
I
2
T
I
3
T
I
6
ð1012Þ1011 (0112)
0111
(1012)
1011
(0112)
0111
(1102)
1101
(1102)
1101
Group I, 4153 41:34
II, 5053 48:44
III, 3470 87:85
Experiments 10.6% 85.6% 3.8%
SFs (A) 0.490 0.138 0.474 0.129 0.101 0.093
SFs (B) 0.357 0.357 0.370 0.370 0.006 0.006
SFs (C) 0.138 0.490 0.129 0.474 0.093 0.101
Generalized SFs One high, One moderate One high, One moderate Two are same and low
DGA 0.01 0.01 0.18 0.18 0.05 0.05
NDD Very difﬁcult Unless dislocations
climb
Energetically favored by
prismatic <a>dislocations
Facilitated by prismatic <a>
dislocations
The bold font indicates positive and greater SFs.
Fig. 10. (a) Illustration of the common line between a prismatic plane, a Group II
twinning plane, a Group III twinning plane and a primary twin plane along ½2423. (b)
Schematic of reaction of a prismatic dislocation at a primary twin boundary into a
Group II or a Group III twin dislocation plus a residual dislocation.
S. Xu et al. / Acta Materialia 132 (2017) 57e68 65
variants T
I
i
are classiﬁed in to 3 groups because of the same SF for a
pair of twins T
I
i
and T
I
iþ3
. A generalized SF analysis in the loading
domain where the two primary twins are mechanically preferred
was performed to select sequential twins. The results show that the
SF analysis made the correct choice for group, but wrong for twin
variant in the group. Displacement gradient accommodation anal
ysis correctly selected the twin variant in the group which is
determined by the SF analysis. When the local stress associated
with twintwin interactions is taken into account for twin nucle
ation, the location of sequential twin is well predicted in the obtuse
region.
5.2. C
I
i
/T
I
j
double twinning
{1122 }/{1012} double twins are also characterized in rolled
pure titanium at room temperature. We examined the selection of
secondary twin variant according to the SF analysis, displacement
gradient accommodation, and nucleation based on dislocation
dissociation. Corresponding to the misorientation between sec
ondary twins and the primary twin, six twin variants T
I
i
are
conventionally classiﬁed into 3 groups dGroup I: C
I
i
/T
I
iþ3
and
T
I
iþ4
, Group II: C
I
i
/T
I
i
and T
I
iþ1
, and Group III: C
I
i
/T
I
iþ2
and T
I
iþ5
.
Such classiﬁcations show the advantage regarding displacement
gradient accommodation, two variants in the same group have the
same contribution to displacement gradient accommodation.
However, such classiﬁcation is inconvenient to the discussion of the
activity of secondary twins with respect to the SF. Twin variants T
I
i
and T
I
iþ3
have the same SF, but they are categorized into Groups I
and II, because experimental observations show a strong correla
tion of twin variant with the SF.
Fig. 11a shows an EBSD pattern where the contraction twin
contains two secondary extension twins. The trace of the twinning
plane associated with the contraction twin is nearly parallel to the
horizontal axis. With the help of pole ﬁgures in Fig. 11b, a misori
entation angle of the ð1212Þtwin plane between C
I
5
twin and the
matrix is 3.35
. Therefore, the contraction twin is determined to be
twin variant C
I
5
. In the pole ﬁgure of {1012} planes (Fig. 11c), a
misorientation angle between ð0112Þplanes in the T
I
5
twin and in
the matrix is 1.21
, and that between ð1102Þplanes in the T
I
6
twin
and in the matrix is 0.80
.Fig. 11d shows the pole ﬁgure of the
corresponding twinning directions. The misorientation angle be
tween ½0111directions in the T
I
5
twin and in the matrix is 1.22
,
and that between ½1101directions in the T
I
6
and in the matrix is
0.81
. According to the crystallography of twins {1012}<1011>, the
two secondary twin variants are twin variants T
I
5
and T
I
6
, belonging
to Group II. We compared experiment result with the SF analysis. As
shown in Fig. 11a, C
I
5
is the primary contraction twin, T
I
5
and T
I
6
are
secondary twin. The SFs associated with C
I
5
,T
I
5
and T
I
6
are 0.4295,
0.3511 and 0.3927, respectively. The SF of T
I
6
is larger than that of T
I
5
and the difference between T
I
5
and T
I
6
is 0.041, which is smaller than
0.05. This is in agreement with our prediction. The loading direc
tion identiﬁed for the grain according to the projection of external
ND into the crystal frame is located at the predicted loading
domain, as indicated by a black dot in Fig. 9e. This implies that the
SF criterion is well satisﬁed. The displacement gradient accom
modation and the nucleation of TDs based on dislocation dissoci
ation are also satisﬁed.
Fig. 12a shows an EBSD pattern where the contraction twin
contains one secondary extension twin variant. With the help of
pole ﬁgures in Fig. 12b, the misorientation angle between ð1122Þ
planes in the C
I
4
twin and in the matrix is 0.85
. Therefore, C
I
4
twin
variant can be identiﬁed as the primary twin. In the pole ﬁgure of
{1012} planes (Fig. 12c), the misorientation angle between ð1012Þ
planes in the T
I
4
twin and in the matrix is 0.72
. The misorientation
angle between ½1011directions in the T
I
4
twin and in the matrix is
0.73
as shown in Fig. 12d. T
I
4
is thus the active twin variant,
belonging to Group II. The same analysis was performed for the
case in Fig. 12. We found that the SFs associated with C
I
4
and T
I
4
are
0.3222 and 0.473. The other twin variant T
I
5
in Group II is 0.202,
which is much smaller than T
I
4
. This explains why it is unfavorable
and only one twin variant appears, and T
I
4
is activated. The loading
direction identiﬁed for the grain according to the projection of
external ND into the crystal frame is located at the predicted
loading domain, as indicated by a blue dot in Fig. 9f, which shows
the loading domain for C
I
4
variant where the SF of C
I
4
variant is
greater than 0.3 and is the greatest among six equivalent variants.
This result again demonstrates the SF criterion must be satisﬁed
and predict the selection of twin variants. The displacement
gradient accommodation and the nucleation of TDs based on
dislocation dissociation are also satisﬁed, but can not predict the
selection of the activated twin variant.
Fig. 11. (a) An EBSD map of two secondary {1012} twins in a primary {1122}
contraction twin as well as pole ﬁgures of (b) {1122} plane, (c) {1012} plane, (d) 1011
direction. In (b), the black dots represent six {1122} twin planes in the matrix, the pink
diamonds represent six {1122} in the twin. In (c), the black dots represent six {1012}
twin planes in the matrix, the red diamonds and blue stars represent six {1012} twin
planes in the two {1012} twins, and (d) the black dots represent six 1011 twin di
rections in the matrix, the red diamonds and blue stars represent six 1011 twin di
rections in the twin. (For interpretation of the references to colour in this ﬁgure legend,
the reader is referred to the web version of this article.)
Fig. 12. (a) An EBSD map of one secondary {1012} twin in a primary {1122} contraction
twin as well as pole ﬁgures of (b) {1122} plane, (c) {1012} plane, (d) 1011 direction. In
(b), the black dots represent six {1122} twin planes in the matrix, the pink diamonds
represent six {1122} in the twin. In (c), the black dots represent six {1012} twin planes
in the matrix, the blue stars represent six {1012} twin planes in the {1012} twin, and
(d) the black dots represent six 1011 twin directions in the matrix, the blue stars
represent six 1011 twin directions in the twin. (For interpretation of the references to
colour in this ﬁgure legend, the reader is referred to the web version of this article.)
S. Xu et al. / Acta Materialia 132 (2017) 57e6866
6. Conclusions
We experimentally studied sequential twinning in rolled pure
titanium at room temperature, and identiﬁed a new sequential
twinning mechanism T
II
i1
T
II
i
/T
I
i
, i.e., {1012} extension twin stim
ulated by the primary {1121} twintwin junction. The SF analysis as
the local stress is taken into account is able to determine the po
sition of sequential twin variant, but cannot determine the twin
variant in the group of twin variants. Displacement gradient ac
commodation is further used to determine the twin variant.
The wellknown secondary {1012} extension twins in primary
{1122} contraction twins (referred to as C
I
i
/T
I
j
double twinning)
are also analyzed according to our EBSD analysis. Experiment ob
servations can be well accounted for according to the SF analysis.
Displacement gradient accommodation and the nucleation of TDs
based on dislocations dissociation only work well for the most
active twin variants, but cannot address other phenomena associ
ated with C
I
i
/T
I
j
double twinning. For example, the three criteria
predict that Group II twins are predominant among three groups, in
consistent with statistical analysis. However, the displacement
gradient accommodation and the nucleation of TDs based on
dislocation dissociation cannot predict the selection of the acti
vated twin variant in Group II. Compared to them, The SF analysis is
able to predict the activated twin variant. For activated twin vari
ants in Group I and III, displacement gradient accommodation was
not satisﬁed, showing the opposite trend.
Acknowledgements
S. Xu thanks the Ministry of National Education, Higher Educa
tion and Research of France for providing the Doctoral Contract.
M.Y. Gong, X.Y. Xie and J. Wang acknowledge the support by the
Nebraska Center for Energy Sciences Research, University of
NebraskaLincoln. Jian Wang also acknowledges support from the
US National Science Foundation (NSF) (CMMI1661686).
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S. Xu et al. / Acta Materialia 132 (2017) 57e6868
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