Biomedical Engineering Reference
In-Depth Information
has been chosen to mirror such characteristics. Herein, the shear modulus
μ
e
as
single material parameter seems sufficient. Further, the first invariant
I
1
=
tr
C
is
defined as the trace of
C
.
5.2.3.2 Collagen
The second main connective tissue component in arteries, collagen (
Ψ
c
), dominates
their mechanical behavior by a stress-stretch relation of exponential type along the
fiber direction. Experimental observations by, e.g., Dahl et al. (
2007
) indicate that
collagen fibers are preferably aligned with the vessels longitudinal axis, helically
and circumferentially. Thus, the dispersion of the collagen fibers by incorporation
angles
Θ
c
,i
has been measured. In doing so, the relative frequency
f
c
has been re-
garded to fulfill the relation
n
c
i
1 with
n
c
being the number of different
directions
i
. For the modeling of such material characteristics, the used strain en-
ergy is weighted in every incorporated direction with the corresponding, measured
collagen fraction
f
c
,i
and can be written as
1
f
c
,i
=
=
n
c
Ψ
c
=
f
c
,i
Ψ
c
,i
.
(5.15)
i
=
1
Herein, the strain-energy functions
c
1
2
c
2
c
2
(λ
c
,i
−
1
)
2
exp
[
]
if
λ
c
,i
≥
1
,
Ψ
c
,i
=
(5.16)
0
else
,
depend on two material constants,
c
1
and
c
2
, see Holzapfel et al. (
2000
).
5.2.3.3 Smooth Muscle Cells
The third component in arteries are mainly circumferentially oriented SMCs. How-
ever, it stands out that there is a certain stretch at which the generated force reaches
a maximum value, see Schmitz and Böl (
2011
). Having those experimentally ob-
tained force-stretch characteristics in mind, the strain-energy function of a single
SMC or a layer of SMCs reads
n
s
Ψ
s
=
f
s
,j
Ψ
s
,j
.
(5.17)
j
=
1
According to the findings of Walmsley and Murphy (
1987
) the active strain-energy
function has been weighted with the SMC volume fractions
f
s
,j
in every incorpo-
rated direction
j
. Further,
n
s
denotes the number of considered directions and the
