Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 190

of the structural gradient. In this case, the same situation prevails because the
structural gradient is still inconsistent with the reflective structural symmetry
required by a plane of mirror symmetry. The only possibility is that the normal to
a plane of material symmetry is perpendicular to the direction of the structural
gradient. Thus, it is concluded that the only linear elastic symmetries permitted in a
material containing a structural gradient are those symmetries characterized by
having all their normals to their planes of mirror or reflective symmetry perpendic-
ular to the structural gradient. The caveat to this conclusion is that the structural
gradient and the material symmetry are at the same structural scale in the material.
Only the three linear elastic symmetries, triclinic, monoclinic, and trigonal, satisfy
the condition that they admit a direction perpendicular to all the normals to their
planes of mirror or reflective symmetry. Trigonal symmetry has the highest sym-
metry of the three symmetries and admits a direction that is not a direction
associated with a normal to a plane of reflective symmetry, nor any projected
component of a normal to a plane of reflective symmetry.
In summary, in a material with a structural gradient, if an RVE may be selected
so that is large enough to adequately average over the microstructure and small
enough to insure that the structural gradient across the RVE is negligible, then it is
not necessary to restrict the material symmetry to accommodate the gradient.
However, in a material with a structural gradient, if an RVE cannot be selected
such that the structural gradient across an adequately sized RVE is negligible, then
it is not necessary to restrict the material symmetry to accommodate the structural
gradient.
7.7 Tensorial Representations of Microstructure
The description and measurement of the microstructure of a material with multiple
distinct constituents is called quantitative stereology (Underwood 1969 ) or texture
analysis (Bunge 1982 ) or, in the case of biological tissues, it becomes part of
histology. The concern here is primarily with the modeling of the material micro-
structure and only secondarily with techniques for its measurement.
It is recognized that the volume fraction of a constituent material is the primary
geometric measure of local material structure in a material with multiple distinct
constituents. This means that in the purely geometric kinematic description of the
arrangement of the microstructure the volume fraction of a constituent material is
the primary parameter in the geometric characterization of the microstructure.
The volume fraction of a constituent in a multiconstituent material does not
provide information on the arrangement or architecture of microstructure of the
multiconstituent material, only information on the volume of the constituent pres-
ent. The second best measure of local material microstructure depends upon the
type of material microstructure being modeled and the objective of the modeler.
One approach to the modeling of material microstructures is to use vectors and
tensors to characterize the microstructural architecture. For example, in theories for