Biomedical Engineering Reference
In-Depth Information
1 Introduction
The physical understanding and modelling of damage and failure in materials
presents a major challenge in various engineering-related disciplines. To give an
example, the modelling of damage effects in anisotropic soft biological tissues is
closely related to the continuous failure of fibres embedded in the ambient matrix
material. Arteries, for instance, can be considered as a composite of an isotropic
ground substance of elastin fibres and a highly anisotropic network of cross-linked
collagen fibrils. Mechanical loading beyond the physiological loading range, e.g.
caused by a surgical intervention such as balloon angioplasty, can significantly reduce
the elastic properties of the artery. These softening phenomena can be attributed to
the continuous overstretch and degradation of particular collagen fibres.
Based on the classical work by Kachanov [
8
], who associated damage effects to
an area reduction of the stress-bearing region, material degradation can be modelled
by means of standard continuum damage formulations, i.e. in a local sense. Up to
now, a large variety of models exist where we refer the reader to classic monographs
and textbooks as, e.g. Krajcinovic and Lemaitre [
9
], or Lemaitre [
10
], to name only
two. However, the assumption of a purely
local
continuum damage may imply the
ill-posedness of the underlying boundary value problem. With regard to numerical
methods such as the finite element method, this may lead to mesh-dependent solu-
tions, a vanishing localised damage zone uponmesh refinement, and hence physically
questionable results.
In order to circumvent these deficiencies, i.e. to
regularise
the problem, several
approaches have been proposed in the literature as, for instance, the concept of
generalised
non-local
continua, where internal length scales are introduced into the
continuum formulation, see the monograph by Eringen [
4
], the article by Aifantis [
1
]
or the contributions in Eringen [
3
] and Rogula [
13
]. A non-local continuum formula-
tion can generally be established by either introducing an integral- or a gradient-type
equation.
In this contribution, we apply a non-local gradient-based damage formulation
within a geometrically non-linear setting allowing for large deformations. Concep-
tually following Dimitrijevic and Hackl [
2
], we enhance the local free energy by
a gradient-term. This term essentially contains the gradient of the non-local dam-
age variable which, itself, is introduced as an additional global field variable. In
order to guarantee the equivalence between the local and non-local damage variable,
a penalisation term is incorporated within the free energy. Based on the principle
of minimum total potential energy, a coupled system of Euler-Lagrange equations,
i.e. the balance of linear momentum and the balance corresponding to the non-local
damage field, is obtained and solved in weak form. As a key aspect, the hyper-
elastic constitutive response at local material point level is governed by a highly
non-linear strain energy, additively composed of an isotropic matrix material and
of an anisotropic contribution related to the fibre-reinforcement. The inelastic con-
stitutive response is represented by a scalar
-damage formulation, where only
the anisotropic elastic part is assumed to damage. The resulting coupled, highly
[
1-
d
]
