Biomedical Engineering Reference
In-Depth Information
Tabl e 2. 2
Table with typical
wall parameter values for the
aorta and iliac arteries
Parameters
Aorta/iliacs
Char. radius R (m)
0.006-0.012 [
133
]
1-2
10
3
[
133
]
Wall thickness h (m)
Wall density
K
(kg/m
3
)
1:1
10
3
[
133
]
10
5
-10
6
[
3
,
14
,
133
]
Young's modulus E (Pa)
10
3
-8
10
3
[
3
,
4
,
14
]
Wall viscosity coef. hC
v
=R (Pa s)
Poisson's ratio
0:5
Tabl e 2. 3
Structure
parameters for Example 5
Parameters
Values for model problem
Shear mod. G (dynes/cm
2
)
0:25
10
6
Timoshenko factor k
1
Viscoelasticity (poise cm)
0:01
Radius R (cm)
0:5
Wall density
s
(g/cm
3
)
1:1
Wall thickness h
s
(cm)
0:1
Young's mod. E(dynes/cm
2
)
0:75
10
6
Poisson's ratio
0:5
Example 5: The Linearly Elastic String Model
We present here a model which has been used by several authors to test numerical
solvers for FSI in blood flow [
8
,
9
,
21
,
85
,
125
,
131
]. This model problem was first
introduced by Formaggia et al. in [
67
]. The structure model for this benchmark
problem is of the form
s
h
@
2
r
@t
2
kGh
@
2
r
@
z
2
R
2
@
3
r
Eh
1
2
r
C
@
z
2
@t
D f:
(2.45)
E
Here G D
2.1
C
/
is the
shear modulus
and k is the
Timoshenko shear correction
factor
. The values of the model parameters used in [
67
]aregiveninTable
2.3
.
Notice that this model can be recovered from the linearly viscoelastic Koiter shell
model (
2.42
) by taking the longitudinal component of displacement to be equal to
zero, and by choosing the following values for the coefficients in (
2.43
), (
2.44
):
Eh
R
2
.1
2
/
;C
1
DkGh; D
2
D;
C
0
D
with all the other coefficients equal to zero. The typical values of the parameters in
this model are given in Table
2.3
[
67
]. The Young's modulus E and viscoelasticity
are smaller than the physiological values. This means that the arterial wall in this
example is rather elastic. The relatively large value of the coefficient in front of the
second-order derivative with respect to z (describing bending rigidity) minimizes
the oscillations that would normally appear in such structures. For the typical
physiological values of these parameters see Table
2.2
.
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