Biomedical Engineering Reference
In-Depth Information
By using the relationships between the Lamé constants and Young's modulus of
elasticity E and Poisson ratio ,givenby(
2.4
), operator
L
el
can be written as:
12R
4
.1
2
/
@
4
C R
4
@
z
C 2R
2
@
z
@
2
2@
2
C
h
3
E
L
el
D
(2.17)
h
3
E
6R
2
.1
2
/
@
z
C
hE
R
2
.1
2
/
:
C
Example 2: The Axially Symmetric Koiter Shell Allowing Both Radial and
Longitudinal Displacement
Here, we assume that nothing in the problem depends on . The problem is axially
symmetric, and the displacement is given by
.t;
z
/ D .
z
.r;
z
/;
r
.t;
z
//:
The linearized change of metric tensor and the linearized change of curvature tensor
are given, respectively, by:
@
z
z
0
0R
r
; R./ D
@
zz
r
0
0
r
:
G./ D
(2.18)
The elastic energy of the problem is given by:
Z
L
0
A
Z
L
0
A
h
3
24
h
2
E
el
./ D
G./ W G./
Rdz
C
R./ W R./
Rdz
:
(2.19)
To define a weak formulation of the problem, introduce the following function
space:
V
c
D H
0
.0;L/ H
0
.0;L/ D
ǚ
.
z
;
r
/ 2 H
1
.0;L/ H
2
.0;L/ W
z
.0/ D
z
.L/ D
r
.0/ D
r
.L/ D 0;@
z
r
.0/ D @
z
r
.L/ D 0g:
Then the weak formulation of the linearly elastic cylindrical Koiter shell is given by
the following: find D .
z
;
r
/ 2 V
c
such that
Z
L
0
A
Z
L
0
A
h
3
24
h
2
G./ W G. /
Rdz
C
R./ W R. /
Rdz
Z
L
D
f
Rdz
;
8 2 V
c
;
(2.20)
0
Here f is the surface density of the force applied to the shell, and
A
is the elasticity
tensor given by (
2.3
).
Search WWH ::
Custom Search