Biomedical Engineering Reference
In-Depth Information
10 Implementation and Application of the New Approach for 4D
Numerical Analysis of Scaffolds
In this section, an example of the new approach for predicting the life-cycle of a hy-
drolytic degradable device, and its implementation in ABAQUS standard is shown,
using the Neo-Hookean material model. This is used to simulate PLA-PCL behavior
for fiber geometry. As commented earlier, this implementation was carried out us-
ing a subroutine UMAT and the PYTHON language. Although Neo-Hookean model
was less accurate than the other models, it is not so complicate to implement, since it
uses only one material parameter
μ
1
. Furthermore, it avoids the violation of the 2nd
Law of Thermodynamics, which happens for the other models with negative val-
ues for the material parameters (
μ
2
and
μ
3
). For this 3D case, the first and second
invariants of deviator part of the left Cauchy-Green deformation tensor are given by:
I
B
=
tr
(B)
(27)
1
/
2
(
tr
B)
2
tr
B
2
1
/
2
II
B
=
−
(28)
FF
T
). The Neo-Hookean compressible
hyper elastic model is given by stored energy function of the form:
where
B
is the deviator stretch tensor (
B
=
1
)
2
W
=
(μ
1
/
2
)(I
B
−
3
)
+
G(J
−
(29)
where
G
is a material constant that depends on the compressibility (
G
0forin-
compressible materials).
J
is the determinant of the deformation gradient (
J
=
=
1
for incompressible materials):
J
=
det
(∂x/∂X)
(30)
where
x
is the current 3D position of a material point and
X
is the reference position
of the same point. Then:
J
−
1
/
3
(∂x/∂X)
F
=
(31)
is the deformation gradient with volume change eliminated. The Cauchy stress ten-
sor for the Neo-Hookean model used in this example is given by:
=
+
−
T
(μ
1
/J )
dev
(B)
2
C(J
1
)I
(32)
where
I
is the 2nd order identity tensor.
The first material parameter is calculated as function of the hydrolytic damage,
μ
1
(d
h
)
, according to a linear regression shown in Fig.
11
.
In this example, a 3D model of a fiber was developed by means of a script in
PYTHON language, using solid and axisymmetric elements, with parabolic inter-
polation functions, as well as with reduced and/or hybrid integration. This script
is run by ABAQUS and the degradation time is required as an input parameter
data (Fig.
13
). The hydrolysis rate of the material (
u
) and the strength of the non-
degraded material (
σ
0
) are initially set in the command lines. The material was con-
sidered nearly incompressible (
G
=
10
−
3
). Then the script calculates the hydrolytic
damage (
d
h
) according to Eq. (
9
), and the material strength (
σ
t
) according to Eq. (
8
),
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