Biomedical Engineering Reference
In-Depth Information
and
•
Stress Couples (Bending Moment)
, which describe the bending moments in terms
of the change of curvature tensor, and are defined by
h
3
24
A
M WD
R
./:
At this point we also introduce the effects of prestress by defining the stress
resultant N
ref
that relates the reference pressure p
ref
with circumferential strain [
46
,
113
,
114
]
2
N
ref
D hR
A
c
2
3
00
h
4
5
A
c
(2.12)
R
h
r
0p
ref
so that the total stress resultant, including the effects of prestress, reads
•
Stress Resultant for a prestressed elastic Koiter shell
h
2
A
h
2
N
ref
:
N D
G
./ C
(2.13)
In what follows, we will be providing more specific details on a few concrete
examples of the general framework described above.
Example 1: The Linearly Elastic Cylindrical Koiter Shell with Radial
Displacement
We present the cylindrical Koiter shell equations without the assumption of axial
symmetry. This means that the displacement can be written as:
.t;
z
;/D .
z
.t;
z
;/;
.t;
z
;/;
r
.t;
z
;//:
However, as is common in the blood flow literature, we will be assuming that the
azimuthal and longitudinal components of the displacement are negligible
0;
z
0, i.e., only the radial component of the displacement is different from zero,
so that:
.t;
z
;/D .0;0;
r
.t;
z
;/;/D .t;
z
;/
e
r
./;
where
e
r
./ is the unit vector pointing in the radial direction. Notice that this does
not mean that the flow is axially symmetric, since the radial displacement is a
function of both and
z
.
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