Biomedical Engineering Reference
In-Depth Information
the reflective symmetry transformation is
R
ðe
2
Þ
given by the second of (
4.4
).
The monoclinic form of the tensor
A
is subject to the transformation
T
A
ðLÞ
¼
R
ðe
2
Þ
A
ðGÞ
½
R
ðe
2
Þ
(4.10)
R
ðe
2
Þ
. Substituting
into (
4.10
) the representation for the monoclinic form for
A
and the representation
for
R
ðe
2
Þ
, one finds that
which follows from the first of (A83) by setting
T
¼
A
and
Q
¼
2
3
2
3
2
3
2
3
A
11
00
0
100
0
A
11
00
0
100
0
4
5
¼
4
5
4
5
4
5
;
A
22
A
23
10
001
A
22
A
23
10
001
0
A
32
A
33
0
A
32
A
33
or
2
3
2
3
A
11
00
0
A
11
0
0
4
5
¼
4
5
:
A
22
A
23
0
A
22
A
23
(4.11)
0
A
32
A
33
0
A
32
A
33
The transformation (
4.10
)or(
4.11
) is thus seen to leave the monoclinic form of
the tensor
A
unchanged by the reflection only if
A
32
¼
A
23
¼
0. It follows then that
the form of the tensor
A
consistent with an orthotropic symmetry characterized by
planes of reflective symmetry normal to the
e
1
and
e
2
base vectors must satisfy the
conditions
A
32
¼
0. It is then possible to show that this restriction also
permits the existence of a third plane of reflective symmetry perpendicular to the
first two. This result for orthotropic symmetry is recorded in Table
4.3
.
A transversely isotropic material is one with a plane of isotropy. A plane of
isotropy is a plane in which every vector is the normal to a plane of reflective
symmetry. This means that the material coefficients appearing in the representation
of
A
for orthotropic symmetry must be unchanged by any reflective symmetry
transformation characterized by any unit vector in a specified plane. Let the plane
be the
e
1
,
e
2
plane and let the unit vectors be
a
A
23
¼
¼
cos
y
e
1
þ
sin
y
e
2
for any and all
; then the reflective symmetry transformations of interest are
R
ðy
12
Þ
given
by the first of (
4.6
). The orthotropic form of the tensor
A
is subject to the
transformation
values of
y
T
A
ðLÞ
¼ R
ðy
12
Þ
A
ðGÞ
½R
ðy
12
Þ
(4.12)
R
ðy
12
Þ
. Substitution
for
R
ðy
12
Þ
and the orthotropic 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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