Biomedical Engineering Reference
In-Depth Information
2
3
2
3
2
3
A
11
00
0
A
11
0
00
A
33
1
0
0
A
11
00
0
A
11
0
00
A
33
4
5
¼
4
5
4
5
0
cos 2
y
sin 2
y
0
sin 2
y
cos 2
y
2
4
3
5
1
0
0
0
cos 2
y
sin 2
y
0
sin 2
y
cos 2
y
or
2
4
3
5
¼
2
4
3
5
;
A
11
0
0
A
11
00
0
A
11
0
00
A
33
A
33
sin
2
2
A
11
cos
2
2
1
0
y þ
y
4
sin 4
yð
A
11
A
33
Þ
1
A
11
sin
2
2
A
33
cos
2
2
0
4
sin 4
yð
A
11
A
33
Þ
y þ
y
(4.15)
>The transformation (
4.14
)or(
4.15
) is thus seen to leave the isotropic form of the
tensor
A
unchanged by the reflection only if
A
11
¼
A
33
. It follows then that the
isotropic form of the tensor
A
consistent with isotropic symmetry characterized by
planes of isotropy in the
e
1
,
e
2
and
e
2
,
e
3
planes must satisfy the conditions
A
11
¼
A
33
.
Actually there are many ways to make the transition from transversely isotropic
symmetry to isotropic symmetry other than the method chosen here. Any plane of
reflective symmetry added to the plane of isotropy of transversely isotropic symme-
try, and not coincident with the plane of isotropy, will lead to isotropic symmetry.
This result for isotropic symmetry is recorded in Table
4.3
. Algebraic procedures
identical with those described above may be used to show that the form of the tensor
A
consistent with cubic symmetry is identical with that for isotropic symmetry.
Problems
4.6.1 Show that the orthotropic form of the tensor
A
given in Table
4.3
is invariant
under the transformation
R
ðe
3
Þ
constructed in Problem 4.4.3.
4.6.2 Show that the isotropic form of the tensor
A
given in Table
4.3
is invariant
under the transformation
R
ðy
13
Þ
constructed in Problem 4.4.3.
4.6.3 Construct a representation for
A
that is invariant under the seven reflective
transformations formed from the set of normals to the planes of reflective
symmetry given in Problem 4.5.2:
k
,
p
,
m
,
n
,
e
1
,
e
2
, and
e
3
. Does the result
coincide with one of the representations already in Table
4.3
? If it does,
please explain.
4.7 The Forms of the Symmetric Six-Dimensional Linear
Transformation C
In this section the definitions of the material symmetries given in Sect.
4.5
are used
in conjunction with the six-dimensional orthogonal transformation (
4.3
)
characterizing a plane of reflective symmetry and transformation law (A162) to
Search WWH ::
Custom Search