Biomedical Engineering Reference
In-Depth Information
⊡
ʵ
11
⊤
ʵ
22
⊣
⊦
ʵ
33
˃
11
=
C
11
C
12
C
13
C
14
C
15
C
16
n
ʵ
12
(13)
n
ʵ
13
n
ʵ
23
n
which can be transformed by vector algebra into
⊡
⊣
⊤
⊦
C
11
C
12
C
13
C
14
C
15
C
16
˃
11
=
ʵ
11
ʵ
22
ʵ
33
ʵ
12
ʵ
13
ʵ
23
n
n
n
n
n
n
n
(14)
Expanding Eq. (
14
) to all load cases finally gives
⊡
⊤
⊡
⊤
⊡
⊤
1
1
1
1
1
1
1
<˃
11
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
C
11
C
12
C
13
C
14
C
15
C
16
⊣
2
⊦
⊣
2
2
2
2
2
2
⊦
⊣
⊦
<˃
11
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
3
3
3
3
3
3
3
<˃
11
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
=
(15)
4
4
4
4
4
4
4
<˃
11
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
5
5
5
5
5
5
5
<˃
11
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
6
6
6
6
6
6
6
<˃
11
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
The first 6 coefficients
C
11
-
C
16
of the generalized stiffness tensor can now be cal-
culated by multiplying this equation with the inverse strain matrix
<
E
>
.This
enables a general notation to calculate all 36 stiffness components.
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
−
1
1
1
1
1
1
1
1
C
i
1
C
i
2
C
i
3
C
i
4
C
i
5
C
i
6
<˃
ij
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
2
2
2
2
2
2
2
<˃
ij
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
3
3
3
3
3
3
3
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
<˃
ij
>
=
(16)
4
4
4
4
4
4
4
<˃
ij
>
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
5
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
<ʵ
<˃
ij
>
6
6
6
6
6
6
6
<ʵ
11
><ʵ
22
><ʵ
33
><ʵ
12
><ʵ
13
><ʵ
23
>
<˃
ij
>
However, the symmetries of the strain and stress tensors demand a symmetric stiffness
tensor with only 21 components remaining. The numerical procedure could cause
small deviations between the symmetric entries. Practically, they can be averaged.
