Biomedical Engineering Reference
In-Depth Information
The residual stresses also affect the distribution of the circumferential stresses
over the undeformed tube thickness
H
, see Fig.
4
b for a physiological (mean) inter-
nal pressure of
p
33 [kPa]. Apart from the discontinuity in the stress distrib-
ution which is due to the different material properties in media and adventitia, we
observe that the stresses are significantly homogenised over the tube thickness and
that their maximum value is dramatically reduced. This effect has a direct impact on
the characteristic pressure range in which inelastic damage processes start to evolve
as a result of high strain energy and stress levels. This effect is studied in more detail
in the subsequent section.
=
13
.
6.2.2 Inelastic Response
Next, we study the degradation of the fibre-reinforced perturbed tube. As we are
dealing with a force-driven boundary value problem, we use an arc-length method in
order to trace the equilibrium path in case of snap-back-related instabilities. A max-
imum inner radial displacement of
u
r
i
5 [mm] is set as a limiting point in the
load-displacement diagram shown in Fig.
5
. Furthermore, we choose an appropriate
set of material parameters to initiate the damage process as soon as the pressure
exceeds the systolic blood pressure of approximately
p
=
1
.
. The results
are indicated in Fig.
5
, where we adjust the pressure plot-range to a maximum value
of 70
≈
20
.
0
[
kPa
]
0 [kPa]. As a consequence, the stiffening due to the residual-stress-induced
contraction is no longer visible but still present. The lower solid black curve rep-
resents the elastic response of the neo-hookean ground substance, the upper solid
black curve represents the elastic response of the fibre-reinforcement material. The
colored lines represent the response of the damaging structure. As only the fibres
are assumed to be affected by the damage, the associated response is always located
between these two black lines representing the elastic response. Upon successive
damage of the fibres, the structural response converges to the response of the undam-
aged neo-Hookean matrix. The curves associated to
c
d
.
kPa
−
1
mm
2
]
in Fig.
5
a-c show a characteristic snap-back behaviour and follow the neo-Hookean
unloading path once the fibres are completely damaged. As an interesting effect,
the incorporation of residual stresses leads to an increase of the peak pressure
before the overall structural response enters the unloading path. This is a direct
consequence of the homogenisation-tendency of the circumferential stresses: as
observed in Fig.
4
b, larger opening angles
={
0
.
5
,
3
.
0
}[
n
entail a larger reduction in the max-
ʱ
imum circumferential stress
. Therefore, higher pressure levels can be sustained
before the strain energy reaches the threshold to initiate the damage evolution.
Particularly Fig.
5
d shows that the structure exhibits almost no degradation for
c
d
={
˃
ʸ
kPa
−
1
mm
2
.
Furthermore, we investigate the effect of the regularisation parameter
c
d
. Gen-
erally, smoother distributions for the non-local damage variable
0
.
5
,
3
.
0
}[
]
and the damage
function
f
d
are obtained for larger values of
c
d
. This is illustrated in Fig.
6
, where the
damage function
f
d
is shown for
ˆ
M
A
ʱ
=
45
.
0
[
deg
]
,
ʱ
=
90
.
0
[
deg
]
and different
=
.
values of
c
d
at a post-peak pressure of
p
40
0 [kPa], i.e. a point within the unloading
