Biomedical Engineering Reference
In-Depth Information
7 Summary
A well-known problem related to
local
continuum damage formulations is the possi-
ble ill-posedness of the underlying boundary value problem. As a consequence, finite
element simulation results turn out to significantly depend on the discretisation.
Typically, this is indicated by a vanishing localised damage zone, which is con-
strained only by the mesh resolution.
One possibility to regularise the problem, is the use of non-local continuum for-
mulations. In this contribution, we present a non-local gradient-enhanced damage
model within a finite strain setting. We implicitly regularise the problem by enhanc-
ing the local free energy with the gradient of the non-local damage variable and
ensure the equivalence between the local and non-local damage field variable by a
penalisation term incorporated within the free energy. The local elastic constitutive
response is represented by a hyperelastic format, additively composed of an isotropic
and an anisotropic part. The inelastic response is governed by a scalar
-damage
approach, affecting the anisotropic elastic part only. As another major aspect of this
contribution, we incorporate residual stresses which are crucial for the modelling
of soft biological tissues such as arteries. The procedure employed here to incorpo-
rate these effects is based on a multiplicative composition of the total deformation
gradient.
We present a biomechanics-related three-dimensional numerical example, i.e. the
simulation of a thick-walled two-layered artery-like tube subjected to internal pres-
sure and residual stresses. The highly non-linear elastic behaviour of the material
is well reflected by the mechanical response of the tube within the physiological
pressure range. For higher pressure values the material softens significantly, deviates
from the elastic path and finally drops towards the elastic response of the undamaged
neo-Hookean ground substance. Physically, this can be interpreted as a continuous
degradation of the fibres contained in an undamaged matrix material. The dam-
age evolution beyond the physiological range results in an increase of the tube's
diameter at the same mean pressure. Consequently, at the expense of a loss in stiff-
ness, these damage effects can be beneficially used in medical surgery to expand an
artery's diameter and thereby restore the blood flow in atherosclerotic blood vessels
for example. As an interesting consequence of the residual stresses and the related
homogenised stress distribution over the tube thickness, we observe an increase of
the peak pressure before the overall structural response enters the unloading path.
In other words, due to the presence of residual stresses, arteries are able to sustain
higher supraphysiological loads. This illustratively shows the beneficial impact of
residual stresses on the overall behaviour of arteries.
It is important to mention that the assumption of an intact matrix and two equally
damaging fibre families provides only a limited view of the real (bio)mechanical
effects at this stage. Therefore, ongoing research work aims at extending the for-
mulation to three global damage variables, including matrix-damage and individual
damage for each fibre family.
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