82
S. Ghosh et al.
0
2
4
6
Number of twins per parent grain
0
0.2
0.4
0.6
0.8
1
Cumulative distribution function
3D-SEVPM
3D-EBSD
(a)
0
0.1
0.2
0.3
0.4
0.5
Normalized twin distance from parent (x/d)
0
0.2
0.4
0.6
0.8
1
Cumulative distribution function
3D-SEVPM
3D-EBSD
(b)
Fig. 16 Comparison of the cumulative distribution function (CDF) of (a) number of twins per
parent grain n and (b) twin distance d from parent centroid in the 3D-SEVM and assembled EBSD
with a size of 250 μm. (Reprinted from: Bagri [29], with permission from Springer)
a stereographic projection along the [001] axis are shown in Fig. 15c, d. For clarity,
the natural logarithm of the frequency of occurrence is plotted in these figures.
The peaks of the distributions in Fig. 15c, d, which appear in small regions of the
plotted areas, occur at (111). The strong peak at twist boundaries (111) indicates
that the majority of 3 boundaries are coherent twins. The population of grain
boundaries in Fig. 15 are measured in units of multiples of a random distribution
(MRD). Values greater than 1 indicate grain boundaries observed more frequently
than those expected in a polycrystalline material with randomly oriented grains [38].
The results can be further validated by comparing the overall distributions in
the 3D-SEVM and assembled EBSD data. The cumulative distribution functions
(CDFs) of number of twins per parent grain and twin distance from parent centroid
are depicted in Fig. 16. Figure 16a implies that about 50% of the parent grains
remain untwinned in both the 3D-SEVM and assembled EBSD. With the exception
of the twin thickness at higher values, the CDFs are in general very good agreement.
The comparison plots of probability distributions in Figs. 14, 15, and 16 validate the
virtual generation method for polycrystalline microstructures containing twins.
3.3 Estimating M-SERVEs for Polycrystalline Microstructure
with Twins
To estimate the optimal size of a microstructure-based SERVE or M-SERVE that
can capture the statistics of the EBSD image data, the CDFs are compared through
the Kolmogorov-Smirnov (KS) test [52]. The KS test quantifies the maximum
absolute difference in the CDFs for the simulated and experimental volumes.
It is a useful tool for seeking the convergence of statistical distributions as a
function of the M-SERVE size. The CDFs of n and d are compared in Fig. 16.
S. Ghosh et al.
0
2
4
6
Number of twins per parent grain
0
0.2
0.4
0.6
0.8
1
Cumulative distribution function
3D-SEVPM
3D-EBSD
(a)
0
0.1
0.2
0.3
0.4
0.5
Normalized twin distance from parent (x/d)
0
0.2
0.4
0.6
0.8
1
Cumulative distribution function
3D-SEVPM
3D-EBSD
(b)
Fig. 16 Comparison of the cumulative distribution function (CDF) of (a) number of twins per
parent grain n and (b) twin distance d from parent centroid in the 3D-SEVM and assembled EBSD
with a size of 250 μm. (Reprinted from: Bagri [29], with permission from Springer)
a stereographic projection along the [001] axis are shown in Fig. 15c, d. For clarity,
the natural logarithm of the frequency of occurrence is plotted in these figures.
The peaks of the distributions in Fig. 15c, d, which appear in small regions of the
plotted areas, occur at (111). The strong peak at twist boundaries (111) indicates
that the majority of 3 boundaries are coherent twins. The population of grain
boundaries in Fig. 15 are measured in units of multiples of a random distribution
(MRD). Values greater than 1 indicate grain boundaries observed more frequently
than those expected in a polycrystalline material with randomly oriented grains [38].
The results can be further validated by comparing the overall distributions in
the 3D-SEVM and assembled EBSD data. The cumulative distribution functions
(CDFs) of number of twins per parent grain and twin distance from parent centroid
are depicted in Fig. 16. Figure 16a implies that about 50% of the parent grains
remain untwinned in both the 3D-SEVM and assembled EBSD. With the exception
of the twin thickness at higher values, the CDFs are in general very good agreement.
The comparison plots of probability distributions in Figs. 14, 15, and 16 validate the
virtual generation method for polycrystalline microstructures containing twins.
3.3 Estimating M-SERVEs for Polycrystalline Microstructure
with Twins
To estimate the optimal size of a microstructure-based SERVE or M-SERVE that
can capture the statistics of the EBSD image data, the CDFs are compared through
the Kolmogorov-Smirnov (KS) test [52]. The KS test quantifies the maximum
absolute difference in the CDFs for the simulated and experimental volumes.
It is a useful tool for seeking the convergence of statistical distributions as a
function of the M-SERVE size. The CDFs of n and d are compared in Fig. 16.