Biomedical Engineering Reference
In-Depth Information
Table 4.4
The forms of the six-dimensional linear transformation
C
for the triclinic, monoclinic,
and orthotropic material symmetries. Note that these forms are not here required to be symmetric,
three of them are and five 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 the six-dimensional linear transformation
2
4
3
5
c
11
c
12
c
13
c
14
c
15
c
16
Triclinic
c
21
^
^
c
22
^
c
23
^
c
24
c
25
^
^
c
26
c
31
^
^
c
32
^
c
33
^
c
34
c
35
^
^
c
36
c
41
c
42
c
43
c
44
c
45
c
46
c
51
^
^
c
52
^
c
53
^
c
54
c
55
^
^
c
56
c
61
^
^
c
62
^
c
63
^
c
64
c
65
^
^
c
66
2
4
3
5
c
11
c
12
c
13
c
14
00
Monoclinic
c
21
^
^
c
22
^
c
23
^
c
24
00
c
31
c
34
00
c
41
c
42
c
43
c
44
00
0000
^
^
c
32
^
c
33
^
c
55
^
^
c
56
0000
c
65
^
^
c
66
2
4
3
5
c
11
c
12
c
13
000
Orthotropic
c
21
c
23
000
c
31
c
32
c
33
000
000
c
44
00
0000
^
^
c
22
^
c
55
^
0
00000
^
c
66
2
4
3
5
Trigonal
c
11
^
c
12
^
c
13
^
c
14
^
0
0
c
12
^
c
22
^
c
23
^
^
c
14
0
0
c
13
c
23
c
33
0
0
0
c
41
c
41
c
44
0
0
0
p
^
0
0
0
0
^
c
44
c
14
p
^
0
0
0
0
c
41
ð^
c
11
^
c
12
Þ
derive the forms of the three-dimensional linear transformation
C
for different
material symmetries. The developments in this section parallel those in the previous
section step for step. The matrices are different, and the results are different, but the
arguments are identical. However, since this topic involves 6
3
matrices, a computational symbolic algebra program (e.g., Maple, Mathematica,
Matlab, MathCad, etc.) is required to make the calculations simple. One should
work through this section with such a program on a computer.
First, since triclinic symmetry has no planes of reflective symmetry, there are no
symmetry restrictions for a triclinic material and the linear transformation
C
is
unrestricted. This conclusion is recorded in Tables
4.4
and 4.5. Monoclinic sym-
metry has exactly one plane of reflective symmetry. This means that the material
6 rather than 3
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