Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 193

Fig. 7.8 The mean intercept
length ellipse as an indicator
of the normals to planes
of mirror symmetry.
( a ) The direction of the
skewed line cannot be a
normal to a plane of mirror
symmetry because the
direction of increasing mean
intercept length has an anti-
mirror symmetry.
( b ) The directions of the
major and minor axes of the
ellipse are normals to planes
of mirror symmetry. From
Cowin and Mehrabadi ( 1989 )

the inclusion volume orientation (Hilliard 1967 ). As pointed out by Odgaard et al .
(1997), for the same microstructural architecture, these different fabric tensor
definitions each lead to representations of the data that are, effectively, the same.
The existence of a mean intercept or fabric ellipsoid for an anisotropic porous
material suggests elastic orthotropic symmetry. To visualize this result, consider an
ellipse, illustrated in Fig. 7.8 , that is one of the three principal planar projections of
the mean intercept length ellipsoid. The planes perpendicular to the major and
minor axes of the ellipse illustrated in Fig. 7.8(b) are planes of mirror symmetry
because the increasing or decreasing direction of the mean intercept length,
indicated by the arrow heads, is the same with respect to either of these planes.
On the other hand, if one selects an arbitrary direction such as that illustrated in
Fig. 7.8(a) , it is easy to see that the selected direction is not a normal direction for a
plane of mirror symmetry because the direction of increasing mean intercept length
is reversed from its appropriate mirror image position. Therefore, there are only two
planes of mirror symmetry associated with the ellipse. Considering the other two
ellipses that are planar projections of the mean intercept ellipsoid, the same
conclusion is reached. Thus, only the three perpendicular principal axes of the
ellipsoids are normals to planes of mirror symmetry. This means that, if the matrix
materials involved are isotropic the material symmetry will be determined only by
the fabric ellipsoid and that the material symmetry will be orthotropy or a greater
symmetry. If two of the principal axes of the mean intercept length ellipsoid were
equal (i.e. an ellipsoid of revolution or, equivalently, a spheroid, either oblate or
prolate), then the elastic symmetry of the material is transversely isotropic. If the
fabric ellipsoid degenerates to a sphere, the elastic symmetry of the material is
isotropic.
The fabric tensor employed here is denoted by F and is related to the mean
intercept length tensor M by F M -1/2 . The positive square root of the inverse of F

is well defined because M is a positive definite symmetric tensor. The principal axes

¼