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3.5.5 Stereology-based Quantification Method Using Tomography Images
If the orientation data can be approximated by a second-order fabric tensor, the general
approach by Kanatani (1984b) that uses observations made on three orthogonal planes
can be used to obtain the required information. For higher order tensors (i.e., fourth-
order), observations on more planes in different orientations are needed to determine
the components of the tensors. Similar computational schemes can be developed.
Kuo, et al., (1998) presented an implementation scheme on the Q570 image analysis
system based on Kanatani’s method. However, the sampling scheme of this implemen-
tation used a square area, which may introduce bias at the corners. It is also platform-
dependent (based on the Q570 system). In addition, Kuo’s method needs to actually cut
the specimens to reveal the surfaces in different orientations. If more than three or-
thogonal orientations are needed, this method becomes more difficult to implement as
more than one statistically equivalent specimen may be needed. The method presented
in this section uses X-ray tomography imaging and virtual cutting technique to conve-
niently obtain the cross-sections in different orientations. The general approach was
developed by Wang et al. (2001b).

3.5.6 Image Interpolation and Analysis
As illustrated in Section 3.3, image interpolation may be required if the spacing be-
tween two adjacent sections is much larger than the in-plane resolution. Following the
procedure illustrated in Section 3.3, images can be interpolated. The interpolation ends
up with 0.3 mm/pixel resolution horizontally and 0.2 mm/pixel vertically.
The stereological quantification of these damage parameters and tensors requires
the use of three representative images acquired from three orthogonal sections (Figure
3.22a) of a specimen. The different components or phases in the image should be sepa-
rated by appropriate thresholds and converted into binary images for performing mea-
surements. The major steps for determining the fabric quantities include the placement
of parallel lines in different orientations on the images, the counting of intersecting
points, and the computation of the mean solid paths or mean void paths (Figure 3.22b).
Detailed algorithms can be found in Kuo et al. (1998). In this example application, Kuo’s
procedures were modified as follows:

1. Use one actually acquired image, and two orthogonal images obtained from
virtual cuttings through the interpolated stack of the images (Figure 3.22a).
2. Account for the different resolutions in the vertical and horizontal directions by
adopting a resolution of the common minimum cofactor of the two resolutions,
in this case 0.1 mm/pixel for the two resolutions of 0.2 mm/pixel and 0.3 mm/
pixel.
3. Use a circular area as the sampling area. A circular area will minimize the effect
resulted from different sampling lengths in different orientations in a rectangular
sampling area (Figure 3.22b).

3.6 Damage Tensor and Quantification Method
Damage tensor is usually defined as the tensorial representation of the area fraction of
cracks or voids to the total area of representative cross-sections. For a representative
sample, by stereology, the average area fraction over the slices in one orientation is the