Biomedical Engineering Reference
In-Depth Information
The required RVE size can be optimized by using a RVE with geometric sym-
metries and periodic or mixed boundary conditions. Periodic boundary conditions
should consequently have a periodic structure. In most cases this implies a complex
generating of such RVE. Stochastic structures can be mirrored at three surfaces for
instance. Disadvantageously, this increases the element size 8-times and enforces
orthotropic behavior additionally. However, orthotropic RVEs and periodic bound-
ary conditions lead to the effective values directly. It is observed, that for stochas-
tic structures periodic boundary conditions show better convergence as well. Good
results can be obtained using this technique, although the continuity of stress and
strain is violated in the anisotropic case. PAHR and ZYSSET [
8
] proposed a set
of mixed boundary conditions with quite similar properties to periodic conditions
in combination with mirrored RVEs. So, it can be argued, that this set leads to a
similar average value as real periodic boundary conditions, but with less elabora-
tion in design. Unfortunately, no additional experiences and best practice advices
concerning convergence of boundary conditions in combination with RVE size exist
in the literature expect of KANIT [
9
,
10
]. So convergence studies are an important
first step. Uniform displacement conditions (KUBC) and uniform stress conditions
(SUBC) are regarded additionally and compared with the PUMBC-boundary condi-
tions of PAHR und ZYSSET. Three tension and three shear load cases are performed
in total. Tables
1
-
3
list all sets of boundary conditions for one tension and one shear
load case, respectively. The remaining four cases are defined analogical. A special
feature is used for the KUBC set. Here, the constraints are continuously distributed
over the surface by the variables x, y, and z.
2.5 Homogenized Anisotropy
The homogenized anisotropic stiffness matrix can be determined by inserting the
average strain
into Eq. (
9
). Six linear independent load cases
have to be simulated to specify the generalized stiffness matrix completely. This leads
to 36 equations in total. Labeling the different load cases with n
<
ʵ
>
and stress
<
˃
>
=
1
,
2
,
3
,
4
,
5
,
6
leads to the following generalized form.
⊡
⊣
˃
11
ʵ
11
⊤
⊦
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
n
n
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
˃
22
ʵ
22
n
n
˃
33
ʵ
33
n
n
˃
12
ʵ
12
=
(12)
n
n
˃
13
ʵ
13
n
n
˃
23
ʵ
23
n
n
n
11
<
˃
>
Considering only the first normal stress
for any load case, the following
inner product is obtained,
