Biomedical Engineering Reference
In-Depth Information
2
3
2
3
2
3
A
11
00
0
A
22
0
00
A
33
cos 2
y
sin 2
y
0
A
11
00
0
A
22
0
00
A
33
4
5
¼
4
5
4
5
sin 2
y
cos 2
y
0
0
0
1
2
3
cos 2
y
sin 2
y
0
4
5
sin 2
y
cos 2
y
0
0
0
1
or
2
3
1
4
sin 4
2
3
A
11
cos
2
2
A
22
sin
2
2
y þ
y
yð
A
11
A
22
Þ
0
A
11
00
0
4
5
4
5
¼
A
22
0
1
4
sin 4
:
A
11
sin
2
2
A
22
cos
2
2
yð
A
11
A
22
Þ
y þ
y
0
00
A
33
0
0
A
33
(4.13)
The transformation (
4.12
)or(
4.13
) is thus seen to leave the transversely
isotropic form of the tensor
A
unchanged by the reflection only if
A
11
¼
A
22
.
It follows then that the transversely isotropic form of the tensor
A
consistent with
transversely isotropic symmetry characterized by a plane of isotropy in the
e
1
,
e
2
plane must satisfy the conditions
A
11
¼
A
22
. This result for transversely isotropic
symmetry is recorded in Table
4.3
. Algebraic procedures identical with those
described above may be used to show that the forms of the tensor
A
consistent
with the trigonal, tetragonal, and hexagonal symmetries are each identical with that
for transversely isotropic symmetry.
Isotropic symmetry is characterized by every direction being the normal to a
plane of reflective symmetry, or equivalently, every plane being a plane of isotropy.
This means that the material coefficients appearing in the representation of
A
for
transversely isotropic symmetry must be unchanged by any reflective symmetry
transformation characterized by any unit vector in any direction. In addition to the
e
1
,
e
2
plane considered as the plane of isotropy for transversely isotropic symmetry,
it is required that the
e
2
,
e
3
plane be a plane of isotropy. The second plane of
isotropy is characterized by the unit vector
a
¼
cos
y
e
2
þ
sin
y
e
3
for any and all
, then the reflective symmetry transformation of interest is
R
ðy
23
Þ
given by
the second of (
4.6
). The transversely isotropic form of the tensor
A
must be
invariant under the transformation
values of
y
T
A
ðLÞ
¼
R
ðy
23
Þ
A
ðGÞ
½
R
ðy
23
Þ
(4.14)
R
ðy
23
Þ
. Substitution
for
R
ðy
23
Þ
and the transversely isotropic form for
A
into this equation, one finds that
which follows from the first of (A83) by setting
T
¼
A
and
Q
¼
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