Biomedical Engineering Reference
In-Depth Information
Fig. 2.5 Continuous solid
subject to volume forces and
external forces
where u represents the displaceme
n
t, b the body forces and C
t
the traction
boundary where the external forces t are applied. By substitution the Lagrangian
functional L can be rewritten as,
Z
Z
e
T
r dX
þ
Z
u
T
b dX
þ
Z
L ¼
1
2
q u
T
u dX
1
2
u
T
t dC
ð
2
:
50
Þ
X
X
X
C
t
and then minimized,
2
4
3
5 dt ¼ 0
d
Z
Z
Z
e
T
r dX
þ
Z
X
u
T
b dX
þ
Z
C
t
t
2
1
2
q u
T
u dX
1
2
u
T
t dC
ð
2
:
51
Þ
t
1
X
X
Moving the variation operator d inside the integrals,
2
3
Z
Z
Z
dX
þ
Z
du
T
b dX
þ
Z
t
2
ffi
dX
1
2
4
5 dt ¼ 0
1
2
dqu
T
u
d e
T
r
du
T
t dC
t
1
X
X
X
C
t
ð
2
:
52
Þ
Since all operations are linear, changing the order of operation does not affect
the result. In the first term of Eq. (
2.52
) the time integral can be moved inside the
spatial integral,
2
4
3
5
dX
dt
¼
1
2
Z
Z
Z
Z
t
2
t
2
ffi
dX
ffi
dt
1
2
dqu
T
u
dqu
T
u
ð
2
:
53
Þ
X
t
1
X
t
1
Using the chain rule of variation and then the scalar property, the integral can
be rewritten as,
