Biomedical Engineering Reference
In-Depth Information
non-linear system of equations is symmetric and can conveniently be solved by
standard incremental-iterative Newton-Raphson schemes or arc-length-based solu-
tion methods without any need for advanced and computationally expensive solution
methods such as a global active-set-search as applied by Liebe et al. [
11
]. Further-
more, the approach proves to be robust, even at high levels of degradation, and allows
to incorporate any suitable scalar-valued damage formulation.
Apart from the highly nonlinear elastic response of soft biological tissues, it is
well-known that the structural design of arteries is characterised by a fibre-reinforced
multi-layered composite subjected to pronounced
residual stresses
. The complex
interaction of material properties with these residual stress effects and the geometry
guarantees the optimal support under different blood pressures within the vessel. As
a further key aspect of this contribution, we therefore incorporate residual stresses
by means of a multiplicative composition of the deformation gradient.
This article is structured as follows: In Sect.
2
, we summarise relevant kinematic
relations for the geometrically non-linear case and the balance equations of the cou-
pled boundary value problem in weak form. In Sect.
3
, we specify the underlying
constitutive equations, containing the isotropic and anisotropic non-linear elastic
and gradient-enhanced free energies, as well as the continuum damage formulation.
In Sect.
4
, we discretise the governing weak forms by means of the finite element
method resulting in a coupled non-linear system of equations. Last, in Sect.
6
,we
apply the model to illustrative three-dimensional inhomogeneous deformation prob-
lems. In order to show the capabilities of the approach with regard to biomechanics-
related problems, we study a force-driven finite element example by means of an
anisotropic artery-like tube subjected to internal pressure and under consideration of
residual stresses. We conclude with a summary and future perspectives in Sect.
7
.
2 Gradient Enhancement of a Continuum
Damage Formulation
This section summarises the essential kinematic relations and presents the governing
coupled balance equations of the boundary value problemon the basis of the principle
of minimum total potential energy. The related variational form provides the basis
for the finite element discretisation described in Sect.
4
.
2.1 Basic Kinematics
Let
x
=
˕
(
X
,
t
)
describe the deformation of the body, which transforms referential
placements
X
∈
B
0
to their spatial counterparts
x
∈
B
t
. Based on this, the deforma-
tion gradient is defined as
F
which transforms infinitesimal referential line
elements d
X
onto their spatial counterparts d
x
. Furthermore, let d
V
and d
=∇
X
˕
v
denote
