Biomedical Engineering Reference
In-Depth Information
Table 4.3
The forms of the three-dimensional linear transformation
A
for the different material
symmetries. Note that these forms are not here required to be symmetric, three of them are and
three of them are not. However symmetry will be required for all of them by different and
specialized physical arguments in subsequent chapters
Type of material symmetry
Form of linear transformation
A
2
4
3
5
Triclinic
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
2
4
3
5
Monoclinic
A
11
00
0
A
22
A
23
0
A
32
A
33
2
4
3
5
Orthotropic
A
11
00
0
A
22
0
00
A
33
2
4
3
5
Hexagonal, trigonal, tetragonal (some crystal classes)
A
11
A
12
0
A
12
A
11
0
0
0
A
33
2
4
3
5
Transversely isotropic, hexagonal, trigonal,
tetragonal (the other crystal classes)
A
11
00
0
A
11
0
00
A
33
2
4
3
5
Isotropic and cubic
A
11
00
0
A
11
0
00
A
11
or
2
3
2
3
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
A
11
A
12
A
13
4
5
¼
4
5
:
A
21
A
22
A
23
(4.9)
A
31
A
32
A
33
The transformation (
4.8
)or(
4.9
) is thus seen to leave the tensor
A
unchanged by
the reflection only if
A
12
¼
A
21
¼
A
13
¼
A
31
¼
0. It follows then that the form of
the tensor
A
consistent with monoclinic symmetry characterized by a plane
of reflective symmetry normal to the
e
1
base vector must satisfy the conditions
A
12
¼
A
21
¼
A
13
¼
A
31
¼
0. This result for monoclinic symmetry is recorded in
Table
4.3
.
Three mutually perpendicular planes of reflective symmetry characterize
orthotropic symmetry, but the third plane is implied by the first two. This means
that the material coefficients appearing in the representation of
A
for monoclinic
symmetry must be unchanged by one more perpendicular reflective symmetry
transformation. Let
e
2
be the normal to the plane of reflective symmetry so that
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