Biomedical Engineering Reference
In-Depth Information
Example 7: A Koiter Shell Model with Prestress
We follow the general description provided in (
2.13
) and calculate the differential
form of the linearly elastic Koiter shell. The only difference with the examples
presented above is in the coefficient multiplying the non-differentiated term, which
will now have an extra term p
ref
=R. Therefore, in Example 1, the linear operator
L
el
given by Eq. (
2.17
) now becomes
@
4
C R
4
@
z
C 2R
2
@
z
@
2
2@
2
C
h
3
E
12R
4
.1
2
/
L
el
D
:
h
3
E
6R
2
.1
2
/
@
z
C
hE
R
2
.1
2
/
C
p
ref
R
C
In Example 2, this gives rise to the following linearly elastic Koiter membrane
equations with prestress:
1
2
z
C
1
R
0
r
hE
K
h R
z
D f
z
;
hE
R.1
2
/
C p
ref
r
(2.50)
hE
R.1
2
/
z
C
K
h R
z
C
R
D f
r
:
In Example 4, the prestress changes the constant C
0
in (
2.41
), which now becomes
h
2
12R
2
/ C
hE
R
2
.1
2
/
.1 C
p
ref
R
:
C
0
D
2.2.2
Elastodynamics of Structures with Finite Thickness
(“Thick Structures”)
The equations modeling elastodynamics of a structure are typically given in terms of
the displacement vector field d D d.t;x/. Vector field d denotes the displacement
from a given reference configuration
S
. We will be assuming that the reference
configuration of the thick structure is given by a straight cylinder of radius R, length
L and thickness H.SeeFig.
2.2
. The elastodynamics equations describe the second
Newton's law of motion
s
@
tt
d
Dr
S
in
S
;t2 .0;T/;
(2.51)
where
s
denotes density of the thick structure, and
S
is the first Piola-Kirchhoff
stress tensor.
To close the system, we need to specify the dependence of
S
on d.The
relationship between
S
and d depends on the material under consideration. In this
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