Variant selection of primary, secondary and tertiary twins in a ...

2050 S. Mu et al. / Acta Materialia 60 (2012) 2043–2053
4.2. Does the accommodation strain control the growth of the
primary extension twins?
During primary extension twinning, three different variants
were nucleated and grew. The orientations and SFs of
all the possible ETW variants of matrix a were calculated.
We deduced that the twin variants corresponding to
observed orientations b, d and c were at2, at5 and at6,
respectively. The SFs of the selected variants were all relatively
high, especially for at2 (b) and at5 (d) with SFs of
0.404 and 0.392, respectively. However, variant b grew to
a considerably greater extent than the other two. Thus
although nucleation appeared to be influenced by the SF,
the latter cannot properly account for the different growth
capabilities displayed by twins of similar SF.
In Mg alloys, once ETWs are nucleated, they grow continuously
during straining and gradually replace the
matrix. In the present case, the twin with the greatest ease
of growth consumed its parent grain almost completely and
thus essentially contacted all the adjacent grains. Once a
twin reaches a grain boundary, the immediate region of
the adjoining grain is sheared and stress concentrations
are accumulated at the boundary. These stress concentrations
are relaxed by slip and twinning in the neighboring
grain, producing what are known as the accommodation
strains. By rotating the twinning shear displacement gradient
tensor from the twin system frame into the crystal
frame of the neighboring grain, the nature of the accommodation
strain that the ETW is trying to impose on the
neighboring grain can be determined. When the growth
of a particular ETW variant requires the operation of a
“difficult” deformation mechanism such as prismatic slip
in the adjacent grain, high back-stresses are generated
and such growth is hindered. Since the prismatic glide
strain components e xy and e yx are classified as “difficult”,
in the following, only these components will be considered
to govern twin variant selection.
At e = 0.11, primary ETW b had already almost completely
consumed its parent grain and contacted most of the
neighboring grains. The accommodation strain tensors of
all six matrix a twin variants in all the adjacent grains were
therefore calculated. However, since the growth competition
only concerned the three selected ETWs at2, at5 and
at6, only the values for these are plotted in Fig. 5. Ase xy
is one of the “difficult” strain components, in Fig. 5, the values
of e xy /e xy (min) are plotted for neighboring grains I–XI
(Fig. 5a), where e xy (min) is the lowest e xy imposed by one
of the six twin variants on the neighboring grain being
considered.
For ease of comparison, the data of Fig. 5a are
regrouped and replotted in Fig. 5b, where the values are
collected for each of the variants. 1 It is clear that variant
1 In some of the neighboring grains, two orientations obeying an ETW
relationship were detected. Both orientations were considered as adjacent
grains, although only the lower value of e xy was used for plotting
purposes.
Fig. 5. (a) Accommodation strains e xy /e xy (min) that three of the primary
extension twins (at2, at5 and at6) are trying to impose on neighboring
grains I–IX; (b) the values of e xy /e xy (min) are collected for each of the twin
variants (at2, at5 and at6). Here e xy (min) is the smallest e xy being imposed
by one of the six variants on the adjacent grain.
at2 (b) requires the lowest e xy in all the neighboring grains
(except for IX), whereas variants at5 (d) and at6 (c) require
much larger amounts of the “difficult” e xy strain (as much
as 10–100 times more, which is not evident due to the curtailment
of the y-axis). Twin variant at6, which corresponds
to twin orientation c, displays relatively low e xy /
e xy (min) values with respect to its adjoining grains, i.e.
grains VI–IX, as shown in Fig. 5b. By contrast, twin variant
at5 (d), which had a higher SF (0.392) than at6 (0.323),
nevertheless did not grow, an observation that can be readily
explained by its relatively high values of e xy in most of
the neighboring grains (see the second set of columns in
Fig. 5b). The values of e xy + e yx were also calculated for
the three variants. These led to similar conclusions and
are not presented here.
4.3. Does the accommodation strain control the nucleation of
the secondary contraction twins?
In this section we test the validity of the accommodation
strain hypothesis by applying it to the selection of the secondary
CTW variants. As shown in the EBSD map of
Fig. 3a, two sets of secondary CTWs e and g, as well as
two sets of tertiary ETWs f and h were detected. It was concluded
above that CTWs e and g were the matrices for tertiary
twins f and h, respectively. Therefore, when we
consider the grains adjacent to the secondary CTWs, the
ones in contact with the tertiary twins should also be taken
into account (as these were formerly neighboring grains of