Biomedical Engineering Reference
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symmetry must be unchanged by any reflective symmetry transformation
characterized by any unit vector in a specified plane. Let the designated plane of
isotropy be the
e
1
,
e
2
plane and let the unit vector be
a
¼
cos
y
e
1
þ
sin
y
e
2
for
any and all values of
; then the reflective symmetry transformation of interest is
R
ðy
12
Þ
, given by the first of (
4.7
). The orthotropic form of the tensor
y
C
must be
invariant under the transformation
T
C
ðLÞ
¼ R
ðy
12
Þ
C
ðGÞ
½R
ðy
12
Þ
;
(4.18)
¼ R
ðy
12
Þ
. The pattern of this
calculation follows the pattern of calculation in (
4.12
). That pattern is the substitu-
tion of the orthotropic form for
Q
which follows from the first of (A162) by setting
C
and
R
ðy
12
Þ
into this equation and the execution
of the matrix multiplication. The resulting matrices are not documented here. They
may be easily obtained with any symbolic algebra program. The result is that the
tensor
C
is unchanged by the reflection only if
c
22
¼ ^
c
11
; ^
c
21
¼ ^
c
12
; ^
c
23
¼ ^
c
13
; ^
c
32
c
12
. It follows then that the form of the tensor
C
consistent with transversely isotropic symmetry characterized by a plane of isot-
ropy whose normal is in
e
3
direction must satisfy the conditions
¼ ^
c
31
; ^
c
55
¼ ^
c
44
; ^
c
66
¼ ^
c
11
^
^
c
22
¼ ^
c
11
; ^
c
21
¼ ^
c
12
; ^
c
23
¼ ^
c
13
; ^
c
32
¼ ^
c
31
; ^
c
55
¼ ^
c
44
; ^
c
66
¼ ^
c
11
^
c
12
. This result
for
trans-
versely isotropic symmetry is recorded in Table
4.5
.
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
C
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 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 vectors
a
, then the reflective
symmetry transformations of interest are
R
ðy
23
Þ
given by the second of (
4.7
). The
form of the tensor
C
representing transversely isotropic symmetry must be invariant
under the transformation
¼
cos
y
e
2
þ
sin
y
e
3
for any and all values of
y
T
C
ðLÞ
R
ðy
23
Þ
C
ðGÞ
R
ðy
23
Þ
¼
½
;
(4.19)
¼ R
ðy
23
Þ
. The pattern of this
calculation follows the pattern of calculation in (
4.14
) and (
4.15
). That pattern is the
substitution of the transversely isotropic form for
C
and
R
ðy
23
Þ
into this equation and
the execution of the matrix multiplication. The resulting matrix is not recorded
here; it may be easily obtained with any symbolic algebra program. The result is
that the tensor
C
is unchanged by any of the reflections whose normals lie in the
Q
which follows from the first of (A162) by setting
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