Biomedical Engineering Reference
In-Depth Information
has been to consider bone as a periodic microstructure from which homogenized
macroscale constitutive parameters are obtained as a function of the microstructure
design. Simple isotropic material with penalization (SIMP) is an approach taken
from topology optimization that was largely used as a first approximation of an
artificial isotropic elastic material with behavior dependent on a single parameter:
an artificial relative density
ρ
∈
[0 : 1]. This model simply reads [43]:
C
ρ
= C
ρ
n
(2.1)
where
C
is the isotropic linear elasticity tensor corresponding to the solid bone. The
ρ
=
ρ
=
bounds
1 are used to represent a void or solid (cortical) bone, respec-
tively. A second common approach is the assumption of a geometrically defined
periodic microstructure from which homogenized properties can be obtained.
Among several proposed alternatives in literature, the present study is carried out
using the periodic microstructure shown in Figure 2.2, proposed by Bagge [18,
20]. Using homogenization techniques, an orthotropic elastic material elasticity
tensor dependent on a relative density
0and
ρ
∈
[0 : 1] related with the thickness of the
microstructure bars is obtained. The microstructure orientation with respect to
global axes is given by the Euler angles
θ
={
θ
1
,
θ
2
,
θ
3
}
T
. No distinction is made
between trabecular and cortical bones. The latter case is assumed to be the limit case
ρ
=
1 of the former one, although it is known that this is just an approximation.
The ''solid material'' is considered to be isotropic linear elastic, with an elasticity
modulus
E
=
υ
=
.
3 [44].
The considered microstructure has cubic symmetry that provides
E
1
20 GPa and Poisson coefficient
0
E
3
,
G
12
=
G
13
=
G
23
,and
υ
12
=
υ
13
=
υ
23
in the material coordinates. The coefficients
of the elasticity tensor are given below [18]:
=
E
2
=
1111
2222
3333
3
2
C
= C
= C
=
5409
.
96
ρ
+
8
.
636
ρ
1122
1133
2233
5
4
C
= C
= C
=
.
ρ
+
.
ρ
938
144
720
29
1212
1313
2323
4
3
C
= C
= C
=
1789
.
34
ρ
+
118
.
038
ρ
(2.2)
y
z
x
{e
x
, e
y
, e
z
}
Y
Z
{e
X
, e
Y
, e
Z
}
X
[
•
]
X
= [R] [
•
]
x
Figure 2.2
Idealized trabecular bone microstructure.
