Biomedical Engineering Reference
In-Depth Information
families of collagen fibers
Ψ
fib
1
and
Ψ
fib
2
, and a contribution of the smooth muscle
cells
Ψ
smc
. The individual contributions will be specified in detail in the following
section.
10.4.1.1 Volumetric Bulk Material
The volumetric free energy
Ψ
vol
can, for example, be expressed as follows Arruda
and Boyce (
1993
):
Λ
1
ln
(J)
,
2
J
2
1
−
Ψ
vol
=
−
(10.2)
where
J
is the determinant of the deformation gradient
F
. The penalty parameter
Λ
corresponds to
κ/
2, with
κ
the bulk modulus (in MPa), and should be set high
enough to ensure a nearly incompressible response.
Since this term is handled separately in an incompressible finite element formu-
lation, we will now focus on the four contributions to the deviatoric energy
Ψ
dev
,
which are the primary descriptors of the material behavior.
10.4.1.2 Damage to the Deviatoric Components
All deviatoric components are allowed to undergo degradation in the case of physio-
logical overload. Simo and Ju (
1987
), in general, and Balzani et al. (
2006
) for arter-
ies have described the approach of weighting the strain energy with a scalar-valued
damage variable
(
1
d)
. This model builds upon the classical damage concept, and
introduces an independent damage variable for each individual constituent. Thus,
−
=
1
d
i
Ψ
i
.
Ψ
i
−
(10.3)
Here,
Ψ
i
denotes the elastic energy of one of the deviatoric constituents (mat, fib
1
,
fib
2
). The smooth muscle cells form an integral part of the matrix constituent, even
in their passive state. Therefore, their degradation is assumed to depend on both the
passive damage
d
smc
pas
in the surrounding matrix and the active damage
d
smc
act
in the
smooth muscle cells themselves:
=
1
pas
1
act
Ψ
smc
.
Ψ
smc
d
smc
d
smc
−
−
(10.4)
The evolution of the damage variable of each constituent
d
i
is driven by the undam-
aged elastic energy, as proposed by Balzani et al. (
2006
):
γ
i
1
exp
−
β
i
/m
i
with
i
d
i
mat
,
fib
1
,
fib
2
,
smc
pas
,
smc
=
−
=
act
.
(10.5)
The weighting factor
γ
i
(in kPa) can be used to tune the sensitivity to damage,
γ
i
∈
, and
m
i
is a dimensionless parameter of the damage model. The variable
β
i
[
0
,
1
]
is
