Biomedical Engineering Reference
In-Depth Information
˃(ʵ
=
)
=
ʻ
=
ʽ/((
+
ʽ)
with the assumption
0
0 and the Lame's constants
E
1
(
−
ʽ))
=
/(
(
+
ʽ))
[
12
].
The physical validity of this equation expires with increasing deformation due to
the linearization and does not describe the stress decrease of real material. However,
in practical (
5
) and (
7
) only differ for deformations higher than 5% strain.
The vectorization of the stress and strain tensors by use of the Voigt notation leads
to the well-known matrix equation:
⊡
⊣
1
2
and
G
E
2
1
⊤
⊦
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
1
−
ʽʽ
ʽ
000
˃
11
˃
22
˃
33
˃
12
˃
13
˃
23
ʵ
11
ʵ
22
ʵ
33
ʵ
12
ʵ
13
ʵ
23
ʽ
1
−
ʽʽ
000
E
ʽ
ʽ
1
−
ʽ
000
=
(8)
1
−
ʽ
2
00
2
0
0
0
(
1
+
ʽ)(
1
−
2
ʽ)
1
−
2
ʽ
0
0
0
0
0
2
1
−
2
ʽ
0
0
0
0
0
2
This linearized equation still describes pure isotropic material behavior. It can
be converted into the generalized Hooke's law by a phenomenological motivated
consideration, so that in principle all 36 coefficients can be chosen independently.
⊡
⊤
⊡
⊤
⊡
⊤
˃
11
˃
22
˃
33
˃
12
˃
13
˃
23
C
11
C
12
C
13
C
14
C
15
C
16
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
ʵ
11
ʵ
22
ʵ
33
ʵ
12
ʵ
13
ʵ
23
⊣
⊦
⊣
⊦
⊣
⊦
=
(9)
However, both the strain tensor
ʵ
and the stress tensor
˃
are symmetric, so the stiffness
C
−
1
have to be symmetric as
well. Consequently, only 21 independent coefficients remain, which have to provide
a positive determinant.
One should consider that this kind of anisotropic modeling is only valid for homo-
geneous bodies. However, anisotropic behavior in general is caused by inhomogene-
ity, so this is a crude assumption with very limited validity. For example, a structure
causing momentums cannot be homogenized with the constitutive law (
9
).
However, the estimated stiffness is strongly influenced by the numerical process
even for suitable structures.
matrix
C
and the respective compliance matrix
N
=
2.3 Homogenization Approach
In general an analogical homogeneous constitutive law for the underlying inhomo-
geneous microstructure should fulfill the Hill condition [
13
].
<
˃
><
ʵ
>
=
<
˃ʵ
>
(10)
