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

2052 S. Mu et al. / Acta Materialia 60 (2012) 2043–2053
of the matrix, which has been virtually completely replaced
by the twin. Two other first-order twins, grains c and d, were
also formed, but occupy much smaller volumes than the
dominant first-order twin b. These first-order twins contain
several second-order twins, such as grains e and g. Grains f
and h are third-order twins that have essentially consumed
their second-order twin parents e and g, respectively.
To investigate the selection of the first-order twin variants,
we performed full constraint Taylor simulations on
all 11 neighbor grains of grain b. For this purpose, the
shape change attributable to the first-order twinning was
imposed on each nearest neighbor grain separately and
the slip (c) distributions on the activated systems were
determined. In this way, it was possible to establish which
variant required the least expenditure of energy (Taylor
energy) P n
s¼1 s sc s , where n is the number of activated slip
and twinning systems and s the critical resolved shear stress
of the respective system. This simulation was carried out
for each of the six first-order twin variants. For calculation
of the Taylor energy, we assumed ratios of the critical
resolved shear stresses s for basal, prismatic and pyramidal
slip, as well as extension twinning of 1:9:10:4 with respect
to basal slip. This has been shown to yield good results
in Mg deformation texture simulations [14].
The results are presented in Fig. 8a in terms of the Taylor
energies required for the formation of the three variants in
each neighboring grain. The total Taylor energies summed
over all 11 neighboring grains are illustrated in Fig. 8b.
(The other three variants, which did not form in the microstructure,
require higher total work than the observed three
variants. Therefore, their Taylor energies are not presented
in the figures.) Although variant c requires the lowest
energy, it also has the lowest SF (0.323). Conversely, variant
d, with a higher SF (0.392), is barely detectable in the microstructure,
apparently because of its high Taylor energy.
Such a first-order twin simulation is only an approximation,
of course, since the adjacent grains do not in fact
deform homogeneously. Furthermore, many nearest neighbor
grains are also affected by grain interactions with their
adjacent grains. The third-order twins, on the other hand,
are only surrounded by a single first-order twin, i.e. firstorder
twin b, see Fig. 3. The Taylor analysis in this case,
with only one neighbor grain, i.e. grain b, yields the results
for twins f and h, illustrated in Fig. 9. Here it can be seen
that variant et5 (i.e. f) of second-order twin e and variant
gt6 (i.e. h) of second-order twin g are energetically preferred,
in excellent agreement with the observations.
From this analysis, we conclude that the accommodation
strain hypothesis is indeed consistent with an energyminimization
principle, as evident from the Taylor simulations
for simple geometries. For more complex arrangements, the
accommodation strain hypothesis (Sections 4.2–4.4) leads
to better predictions than the full constraint Taylor model
of Section 4.5, which does not take the full effects of grain
interaction into account.
Fig. 8. (a) Simulated Taylor energies expended in the 11 nearest-neighbor
grains during the formation of the observed three primary extension twins.
(b) Simulated total Taylor energies associated with the formation of the
three variants normalized to take into account the lengths of the respective
grain interfaces.
Fig. 9. Normalized Taylor energies expended in neighboring grain b
during the formation of tertiary extension twins (a) e ! f and (b) g ! h.
The variants with the lowest Taylor energies were the ones observed in the
experiments.