Biomedical Engineering Reference
In-Depth Information
6.4.2 Structurally motivated constitutive models - damage regime
Motivated by a need to model failure of the IEL separately from failure of the other
components of the arterial wall, a nonlinear, inelastic, isotropic, dual-mechanism
constitutive equation was developed for cerebral artery tissue damage [119]. This
model was subsequently extended to include the collagen derived anisotropy [71],
subfailure isotropic damage to the IEL [74] and collagen damage [70]. While a scalar
damage function was used for collagen in this later work, the resulting damage was
anisotropic. Both discontinuous and continuous damage were considered as well as
enzymatic damage. Balzani et al. previously considered discontinuous damage to
the anisotropic collagen fibres using a scalar damage function [5]. Damage to the
isotropic mechanism was not considered. A method was used to include the effect
of the residual stresses in the unloaded configuration. Circumferential overstretching
of atherosclerotic arteries was modelled.
In this section, we briefly provide the theoretical background for the model. In the
next section, we apply this model to the analysis of damage in arteries. Continuum
damage models are introduced to model the progressive deterioration in mechanical
properties on a scale where it is suitable to homogenize the individual cracks. We
restrict attention to isothermal theory and by way of example consider materials for
which the undamaged response is hyperelastic. It is straightforward to generalize
this approach to rate-type models. In the following discussion, a single mechanism is
considered. Similar approaches follow directly for multi-mechanisms materials [70,
74] and multi-mechanism damage will be used in Sect. 6.5. In this work, attention is
confined to isotropic damage where a scalar damage variable can be used. This can
be extended to anisotropic damage.
Following earlier work on continuum damage mechanics, (e.g. [107]), we begin
by postulating the existence of a Helmholtz strain energy function per unit volume
in
0
(
W
) that depends on
C
0
as well as a scalar damage variable (
d
)Wetakeastrain
space based approach, assuming the dependence on
d
can be explicitly written as,
κ
W
o
=(
−
)
(
C
0
)
W
1
d
(6.41)
where
W
o
is the effective strain energy of the hypothetical undamaged material sub-
ject to the conditions
W
o
W
o
(
I
0
)=
0
,
(
C
0
)
≥
0
.
(6.42)
The internal variable
d
is defined to be in the range
[
0
,
1
]
with zero corresponding to
no damage and one to total damage. The factor
is called the
reduction fac-
tor
(for obvious reasons). This form was first proposed by Kachanov [62] to model
creep rupture of metals. The Clausius-Planck inequality is imposed by requiring the
internal dissipation
(
1
−
d
)
D
in
be non-negative for all times and material points in the body,
W
D
in
=
−
+
σ
:
D
≥
0
(6.43)
