Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 75

symmetries. Some background material is provided before the derivation of
representations. First, there is a discussion of crystalline and textured materials in
Sect. 4.3 . Next the only symmetry operation considered here, the plane of mirror
symmetry, is introduced in Sect. 4.4 , and the symmetries of interest are defined in
terms of mirror symmetry planes in Sect. 4.5 . The symmetry representations for the
forms of A and C associated with the symmetries of interest are then obtained in
Sects. 4.6 and 4.7 .
The primary interest here is only in about half of the eight material symmetries
admitted by the tensor C , however all of them are described for completeness (and
because being complete requires little space). The eight material symmetries
admitted by the tensor C are triclinic, monoclinic, trigonal, tetragonal, orthotropic,
transversely isotropic (or hexagonal), cubic, and isotropic symmetry. The main
interest will be in the orthotropic, transversely isotropic, and isotropic symmetries
with lesser interest in the triclinic, monoclinic, and trigonal symmetries. Curvilinear
and rectilinear anisotropy are described and compared in Sect. 4.8 . The representa-
tion of the symmetry of a material with chirality (handedness) is considered in
Sect. 4.9 . Section 4.10 is a short guide to the literature on the subject matter of this
chapter.

4.2 The Representative Volume Element

The RVE is a very important conceptual tool for forming continuum models of
materials and for establishing restrictions that might be necessary for a continuum
model to be applicable. An RVE for a continuum particle X is a statistically
homogeneous representative of the material in the neighborhood of X , that is to
say a material volume surrounding X . For purposes of this discussion the RVE is
taken to be a cube of side length L RVE ; it could be any shape, but it is necessary that
it has a characteristic length scale. An RVE is shown in Fig. 4.1 ; it is a homogenized
or average image of a real material volume. Since the RVE image of the material
object O averages over the small holes and heterogeneous microstructures, overall
it replaces a discontinuous real material object by a smooth continuum model O of
the object. The RVE for the representation of a domain of a porous medium by a
continuum point is shown in Fig. 4.2 . The RVE is necessary in continuum models
for all materials; the main question is how large must the length scale L RVE be to
obtain a reasonable continuum model. The smaller the value of L RVE the better; in
general the value of L RVE should be much less than the characteristic dimension L P
of the problem being modeled. On the other hand the L RVE should be much larger
than the largest characteristic microstructural dimension L M of the material being
modeled, thus L P

L M . In wood, for example, this can be a problem
because wood has large microstructures and some objects made of wood are small.
For low carbon structural steel the bounds on L RVE are much less restrictive, the
characteristic size of the problem is greater and the characteristic size of the

L RVE