Biomedical Engineering Reference
In-Depth Information
(a)
(b)
Fig. 2
Geometry and finite element mesh of the thick-walled double-layered tube, where only one
quarter is shown.
a
Perfect tube.
b
Perturbed tube
the artery cause the ring to spring open to form an open sector. We now carry out a
numerical reproduction of the opening angle experiment for a thick-walled double-
layered tube, in order to assess whether the method provided in Sect.
5
yields accurate
results. This technique allows for the inclusion of different opening angles for the
different arterial layers within only one simulation. For this purpose, it is neces-
sary to create an initial deformation gradient
F
res
at each Gauss-point of a related
finite element mesh. Note that
F
res
must be inhomogeneously distributed as also the
residual stresses are. This deformation gradient field has to be provided in terms of
the underlying coordinate system used by the finite element code which is typically
described by a Cartesian basis. The procedure to calculate the initial residual strain
field by means of the deformation gradient tensor
F
res
is described in detail in, for
instance, Waffenschmidt [
14
].
The geometrical setting essentially reflects the geometry of a healthy coronary
artery. We use two different geometries, i.e. (i) a perfect tube with constant thickness
and (ii) a perturbed tube with variable thickness, see Fig.
2
. The perturbed tube is
represented by two different cross-sections. A first cross-sections (length
L
1
) with
constant inner radius
R
i
,
1
and a second cross-sections with different inner radii
R
i
,
1
and
R
i
,
2
. The outer radius
R
o
remains constant over the whole length. The functional
relation of
R
i
for
Z
∈[
L
1
,
L
2
]
is provided by
(
Z
904
Z
5
913
Z
4
756
Z
3
259
Z
2
317
Z
R
i
)
=−
0
.
+
3
.
−
5
.
+
3
.
−
0
.
+
0
.
006
,
(48)
Z
L
2
such that
R
i
R
i
,
1
and
R
i
where
=[
Z
−
L
1
]
/
(
Z
=
L
1
)
=
(
Z
=
L
1
+
R
i
,
2
. The geometrical, structural and material parameters for both geometries
are summarised in Table
2
. We choose four different sets of the opening angles
ʱ
L
2
)
=
n
, whereas we neglect the influence of the axial residual stretch at this stage, i.e.
ʻ
z
=
sin
ʲ
n
e
Z
±
0. The initial orientations of the fibres are assumed as
a
01
,
2
=
1
.
