Biomedical Engineering Reference
In-Depth Information
4.4.5. Show that the reflections R ðy 23 Þ and R ðy 23 Þ when evaluated at
y ¼
0 coincide
R ðe 2 Þ , and when evaluated at
with the reflections R ðe 2 Þ and
y ¼ p
/2, they
coincide with the reflections R ðe 3 Þ and R ðe 3 Þ .
4.4.6. Construct the three- and six-dimensional reflective transformations for each
of four normal vectors, k
¼
(1/2)( e 1 þ
(
3) e 2 ), p
¼
(1/2)(
e 1 þ
(
3) e 2 ),
m
3) e 1 - e 2 ). Show that the set of six
vectors k , p , m , n , e 1 , and e 2 form a set that makes a pattern. The pattern
is such that each vector of the set of six vector points is one of six different
directions and makes angles that are each multiples of
¼
(1/2)(
3) e 1 þ
e 2 ), n
¼
(1/2)
l
(
p
/6 with the
other vectors.
4.5 Characterization of Material Symmetries
by Planes of Symmetry
In this section the number and orientation of the planes of reflective symmetry
possessed by each linear elastic material symmetry will be used to define it. These
material symmetries include isotropic symmetry and the seven anisotropic
symmetries, triclinic, monoclinic, trigonal, orthotropic, hexagonal (transversely
isotropy), tetragonal, and cubic. These symmetries may be classified strictly on
the basis of the number and orientation of their planes of mirror symmetry.
Figure 4.10 illustrates the relationship between the various symmetries; it is
organized such that the lesser symmetries are at the upper left and as one moves
to the lower right one sees crystal systems with greater and greater symmetry. The
number of planes of symmetry for each material symmetry is given in Table 4.1
and, relative to a selected reference coordinate system, the normals to the planes of
symmetry for each material symmetry are specified in Table 4.2 . Triclinic symme-
try has no planes of reflective symmetry so there are no symmetry restrictions for a
triclinic material. Monoclinic symmetry has exactly one plane of reflective symme-
try. Trigonal symmetry has three planes of symmetry whose normals all lie in the
same plane and make angles of 120 with each other; its threefold character stems
from this relative orientation of its planes of symmetry. Orthotropic symmetry has
three mutually perpendicular planes of reflective symmetry, but the existence of the
third plane is implied by the first two. That is to say, if there exist two perpendicular
planes of reflective symmetry, there will automatically be a third one perpendicular
to both of the first two. Tetragonal symmetry has the five planes of symmetry
(
a 5 ) illustrated in Fig. 4.11 ; four of the five planes of symmetry have normals
that all lie in the same plane and make angles of 45 with each other; its fourfold
character stems from this relative orientation of its planes of symmetry. The fifth
plane of symmetry is the plane containing the normals to the other four planes of
symmetry. Hexagonal symmetry has seven planes of symmetry; six of the seven
planes of symmetry have normals that all lie in the same plane and make angles of
60 with each other; its sixfold character stems from this relative orientation of its
a 1 -
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