Biomedical Engineering Reference
In-Depth Information
could also be set to zero, and
m
smc
to an arbitrary value. The two remaining param-
act
eters
γ
smc
pas
and
m
smc
pas
were then calibrated using the experimental data, as described
in Famaey et al. (
2012b
). For a systematic approach to calibrate damage material
parameters in a heterogeneous setting, the reader is referred to Mahnken and Kuhl
(
1999
).
10.5 Finite Element Simulation
10.5.1 Arterial Clamping
With the material model described in Sect.
10.4
, it becomes possible to simulate
the experimental process described in Sect.
10.3
. A three-dimensional finite ele-
ment model was built in Abaqus/Standard 6
.
10
2. Here, an idealized cylindrical
geometry was used with an outer radius of 0.58 mm, a wall thickness of 0.14 mm
and an initial length of 0.1 mm. These values were obtained from measurements
on rat abdominal arteries described in Famaey et al. (
2012a
). Hexahedral
C3D8H
elements were used. Because of severe bending, six elements were taken across the
thickness, and seeding in other dimensions was chosen to ensure regular elements.
The numerical implementation of the arterial clamping is subdivided into two steps:
(i) the setting of the initial damage level, (ii) the clamping process itself. Figure
10.3
shows all steps of the clamping simulation.
In the first part, an opened cylindrical segment with an opening angle of 60
◦
is
closed to account for the circumferential residual strains (Balzani et al.,
2007
). Next,
the segment is longitudinally stretched by 50 %, to account for residual strains in
the longitudinal direction. The values for the residual strains were obtained from
experiments described in Famaey et al. (
2012a
). In the third step, the segment is in-
flated to an internal pressure of 16 kPa, which corresponds to physiological systolic
blood pressure. The material model used in this step is the three-constituent damage
model, as described above, however, without accumulation of damage. At the end
of the third step, the undamaged elastic strain energy of each of the four constituents
is written into a matrix of internal or 'solution dependent variables' for each inte-
gration point, using Python scripting. These are the initial damage threshold levels
Ψ
0
, described in Eq. (
10.6
), which will be used in step 4.
Step 4 starts with a new input file, in which the state of the artery after the first
three steps is imported. By importing, the deformations are included as 'initial val-
ues' for the model. The solution dependent variables defined above contain the dam-
age threshold levels
Ψ
0
specified as 'initial conditions' in the input file. The material
model is now updated to enable damage accumulation,
γ
i
>
0, and four extra solu-
tion dependent variables, representing the
β
i
described in Eq. (
10.6
) are added. In
addition, two extra parts are added to the assembly of the system, namely an upper
and lower clamp, which are gradually moved towards each other during step 4, until
a clamping force of 5 mN is reached. A friction coefficient of
μ
clamp
−
0
.
3isused
between the clamp and the outer arterial surface. Finally, also the internal pressure
=
