Biomedical Engineering Reference
In-Depth Information
an internal variable keeping track of the maximum elastic strain energy experienced
so far, within the time interval 0
≤
t
≤
τ
(Balzani et al.,
2006
):
τ
Ψ
i
(t)
Ψ
0
with
i
β
i
mat
,
fib
1
,
fib
2
,
smc
pas
,
smc
=
max
0
−
=
act
.
(10.6)
≤
t
≤
For the specific case of the smooth muscle cell contribution
Ψ
smc
Ψ
mat
=
and
pas
Ψ
smc
act
=
Ψ
smc
.
Since it can be assumed that no damage occurs in the physiological range, the
damage threshold
Ψ
0
is initialized with the strain energy in the respective constituent
at systolic pressure. For heterogeneous problems,
Ψ
0
may, therefore, differ for each
material point, and is thus not strictly a material property.
10.4.1.3 Elastic Energy of the Deviatoric Constituents
The elastic energy of the extracellular matrix is described by a neo-Hookean contri-
bution:
1
2
c( I
1
−
Ψ
mat
=
3
),
(10.7)
I
1
is the first invariant of
where
c>
0 characterizes the matrix stiffness (in kPa) and
the modified right Cauchy-Green tensor.
10.4.1.4 Collagen Fibers
Collagen fibers will only contribute when under tension, where the free energy con-
tributions of the two families of collagen fibers are formulated according to (Gasser
et al.,
2006
)
2
k
2
exp
k
2
I
fib
1
2
−
1
.
k
1
Ψ
fib
1
,
2
=
−
(10.8)
4
,
6
Here,
k
1
>
0 characterizes the fiber stiffness (in kPa) and
k
2
>
0 is a dimension-
less parameter, while
I
fib
4
,
6
are pseudo-invariants for each of the two fiber families,
accounting for fiber dispersion:
I
fib
4
,
6
κ I
1
+
3
κ)I
fib
=
(
1
−
4
,
6
,
(10.9)
with
I
4
and
I
6
the anisotropic invariants characterizing the stretches along the fiber
directions:
I
fib
λ
θ
cos
2
α
fib
1
,
2
λ
z
sin
2
α
fib
1
,
2
.
4
,
6
=
+
(10.10)
Here,
λ
θ
and
λ
z
are the stretches in the circumferential and axial directions, re-
spectively, and
α
fib
1
and
α
fib
2
denote the angles between the circumference and the
mean directions of the fiber families. In the case of arteries, two fiber families are
oriented symmetrically with respect to the cylinder axis, so that
α
fib
1
α
fib
2
=−
and,
I
fib
4
=
I
fib
6
consequently,
.
