Biomedical Engineering Reference
In-Depth Information
2.6 Effective Isotropy
The elastic constants of a respective isotropy can be obtained with a Voigt or Reuss
approximation [
16
].
1
1
=
,
=
30
(
−
)
K
Vo i g t
9
tr
C
G
Vo i g t
3tr
V
tr
C
=
C
11
+
C
22
+
C
33
+
(
C
12
+
C
13
+
C
23
)
tr
C
2
tr
V
=
C
11
+
C
22
+
C
33
+
2
(
C
44
+
C
55
+
C
66
)
(17)
1
15
K
Reuss
=
tr
K
,
G
Reuss
=
2
(
3tr
R
−
tr
K
)
tr
K
=
N
11
+
N
22
+
N
33
+
2
(
N
12
+
N
13
+
N
23
)
tr
R
=
N
11
+
N
22
+
N
33
+
2
(
N
44
+
N
55
+
N
66
)
In theory, similar results are obtained for both approximations if the Hill condition
is fulfilled. However, as described above this cannot be assumed. The advantage of
these special approximations is obvious. The effective solution has to be in between
those two bounds. The approximations give an ultimate upper and lower estimate
of the according moduli, respectively. In practices, they can be used to calculate
effective moduli by determining the arithmetic means of both estimates.
K
Vo i g t
+
K
Reuss
G
Vo i g t
+
G
Reuss
K
effektive
=
,
G
effektiv
=
(18)
2
2
The isotropic stiffness tensor can now be determined by these moduli compliant with
Eq. (
8
). Thereby, the Young's modulus and Poisson ratio are obtained by the bulk
and shear moduli as in the following
9
KG
3
K
3
K
−
2
G
E
=
G
,
ʽ
=
(19)
+
2
(
3
K
+
G
)
The evidence of isotropy can be checked by the Euclidean norm [
8
,
17
,
18
].
C
isotrop
−
C
anisotrp
E
C
isotrop
E
e
=
(20)
This value gives a measure of the error and thus
e
0 means perfect isotropy.
The determined isotropic elasticity moduli are still depending of the RVE random-
ness. AMonte-Carlo simulation is performed in order to obtain the desired accuracy.
Thereby, the average elasticity modulus of a sufficient amount of RVEs is determined
and checked for convergence.
=
