Biomedical Engineering Reference
In-Depth Information
Chapter 4
Modeling Material Symmetry
“A study of the symmetry of three-dimensional spaces is of great theoretical and
practical significance, because symmetrical spaces include crystals (from which, of
course, the majority of solids are formed), and all homogeneous fields without
exception: electric, magnetic, gravitational, etc. A study of the structures of crystals
is unthinkable without a knowledge of the laws governing symmetry of three-
dimensional spaces.” (Shubnikov and Koptsik
1974
)
4.1
Introduction
The variation of material properties with respect to direction at a fixed point in a
material is called
material symmetry
. If the material properties are the same in all
directions, the properties are said to be
isotropic
. If the material properties are not
isotropic, they are said to be
anisotropic
. The type of material anisotropy generally
depends upon the size of the representative volume element (RVE). The RVE is the
key concept in modeling material microstructure for inclusion in a continuum
model. An RVE for a volume surrounding a point in a material is a statistically
homogeneous representative of the material in the neighborhood of the point. The
RVE concept, described briefly in the following section, is employed in this
chapter, which addresses the modeling of material symmetry and, more exten-
sively, in Chapter
7
, which addresses the modeling of material microstructure.
The tensors that appear in linear transformations, for example
A
in the three-
dimensional linear transformation
r
C
in the six-dimensional
¼
A
t
, (A39), and
J
, (A160), often represent anisotropic material
properties. Several examples of these linear transformations as constitutive
equations will be developed in the next chapter. The purpose of this chapter is to
present and record representations of
A
and
C
that represent the effects of material
symmetry. These results are recorded in Tables
4.3
for
A
and Tables
4.4
and
4.5
for
C
, respectively. In these tables the forms of
A
and
T
¼ C
linear transformation
C
are given for all eight
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