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

2044 S. Mu et al. / Acta Materialia 60 (2012) 2043–2053
that neutralize or offset the macroscopic stress and in this
way prevent potential twins with high macroscopic SFs
from forming. Conversely, when accommodation is easy
[1,4], the induced internal stresses are small and even low
macroscopic SFs are sufficient to produce twins. These
concepts have been employed in crystal level models to
account for variant selection during twinning. Therefore,
while detailed quantitative understanding is still lacking,
in part because of the inhomogeneous nature of the localized
strain fields, it has been possible to model some of
the effects of these fields [6–8].
The accommodation strains induced by twinning are
generally relaxed by slip or twinning in the adjoining grain
[9,10]. These strains can be evaluated by rotating the twinning
shear displacement gradient tensor into the crystallographic
frame of the neighboring grain. The analysis of this
phenomenon involves three sets of reference frames: (i) the
twinning system frame; (ii) the crystallographic frame of
the matrix; and (iii) the crystallographic frame of the neighboring
grain. Details of the manipulations required are
described in Ref. [4].
Each of the accommodation strain tensor components e ij
(where i is the shear direction and j is the shear plane) can in
turn be given a physical interpretation in terms of the dislocation
or twinning mechanism involved: (i) the e zx and e zy
components correspond to accommodation by means of
twinning or pyramidal hc + ai slip (here y is parallel to
one of the basal glide Burgers vectors, i.e. to the ½1 1 20Š
direction; z is perpendicular to the basal plane; and x is
mutually perpendicular to both, following the right-hand
rule); (ii) the e yz and e xz shears can be respectively accommodated
by single or double basal glide; (iii) the e yx correspond
to single prismatic glide and the e xy correspond to
balanced glide on the two remaining prism planes. (Note
that the present reference frame differs slightly from that
employed in Ref. [4], in that it has been rotated by 90° about
the z-axis.)
Among the five possible deformation modes (three slip
and two twinning), basal slip and extension twinning can
be activated with relative ease due to their low associated
critical resolved shear stresses. As a result, the magnitude
of the induced back or internal stress is small and there is
little modification of the macroscopic SF. By contrast, prismatic
slip has a critical resolved shear stress (CRSS) at least
an order of magnitude higher than those of basal slip and
extension twinning at ambient temperatures [11]. Moreover,
no alternative mechanism is available to provide the
deformation components (e xy and e yx ) produced by prismatic
slip. In this case, very large localized back-stresses will
be developed, which will tend to oppose the macroscopic
SF. Therefore, when a potential twin variant requires a
large amount of prismatic glide in the neighboring grain,
the nucleation of this variant will be rather difficult even if
the nominal SF is relatively high [4].
The large differences in the energy required to accommodate
the various potential twin variants have been employed
to explain non-Schmid variant selection during secondary
extension [1] and primary contraction [4] twinning in Mg
alloys AZ31 and AM30. In the current work, we extend this
type of variant selection analysis to the nucleation and
growth of primary ETWs in Mg alloys as well as to that
of the two generations of twins that follow. That is, we test
the accommodation strain hypothesis on three further types
of twins in addition to those of Refs. [1,4]. We therefore
consider further the relative importance of the macroscopic
Schmid factor on the one hand and the accommodation
strain (and induced internal stress) on the other.
In the current investigation, samples of Mg alloy AZ31
were deformed in plane strain compression (PSC) to strains
in the range between 0.02 and 0.2 at 100 °C. Most of
the grains were initially favorably oriented for extension
twinning. Copious ETWs were formed on loading and progressively
replaced the starting microstructure by the time a
strain of 0.08 was attained. As deformation was extended
beyond 0.08, many secondary CTWs formed within the primary
ETWs. At a strain of about 0.10, some tertiary
ETWs were generated inside the secondary CTWs. Finally,
when strains beyond 0.13 were applied, the CTWs were
almost completely replaced by the tertiary twins. The orientations
of all three generations of twins were determined
by electron backscatter diffraction (EBSD) techniques.
In what follows, the results obtained at a strain of 0.11
are examined in detail. This strain was chosen because all
three generations of twins could be readily distinguished.
The orientations and SFs of both the observed twins as
well as all potential twin variants were calculated. Both
the values of the macroscopic Schmid factors and the
accommodation strain hypothesis were then employed to
account for the variant selection observed during the nucleation
and growth of the three generations of twins. In order
to calculate the accommodation strains that the twin variants
were trying to impose on their neighboring grains, the
orientations of all adjacent grains were also measured.
Here, only the neighbors observed in the 2-D sections were
considered and the neighbors located above and below the
plane of section were not taken into account. Nevertheless,
new methods, both destructive and non-destructive, for
measuring aspects of 3-D microstructures in bulk polycrystals
are becoming more common. Some of these methods
allow for more direct measurement of aspects of twin formation
[12,13].
2. Experimental
The as-received material was the commercial wrought
Mg alloy AZ31B (3.33 wt.% Al and 1 wt.% Zn) in the
as-cast condition. The ingot was first rolled at 400 °C for
several passes up to 50% thickness reduction. It was then
annealed at 400 °C for 12 h to produce an appropriate
microstructure and the desired initial texture. The resulting
material was fully recrystallized with an average grain size
of about 25 lm, free of mechanical twins. The texture after
annealing was measured by conventional X-ray techniques.
The (0002) pole figure (PF) exhibited a well-defined basal