Biomedical Engineering Reference
In-Depth Information
and in matrix notation, respectively
2
4
3
5
h
e
i
e
j
i
G
11
G
12
G
13
½¼
G
12
G
22
G
23
ð
3
:
71
Þ
G
13
G
23
G
33
with six independent coordinates
"
#
2
2
2
G
11
¼
o
u
1
oX
1
þ
1
2
ou
1
oX
1
þ
o
u
2
oX
1
þ
o
u
3
oX
1
"
#
2
2
2
G
22
¼
o
u
2
oX
2
þ
1
2
ou
1
oX
2
þ
o
u
2
oX
2
þ
o
u
3
oX
2
ð
3
:
72
Þ
"
#
2
2
2
G
33
¼
o
u
3
oX
3
þ
1
2
ou
1
oX
3
þ
o
u
2
oX
3
þ
o
u
3
oX
3
þ
1
2
G
12
¼
G
21
¼
1
2
ou
1
oX
2
þ
o
u
2
oX
1
ou
1
oX
1
ou
1
oX
2
þ
o
u
2
oX
1
ou
2
oX
2
þ
o
u
3
oX
1
ou
3
oX
2
þ
1
2
G
13
¼
G
31
¼
1
2
ou
1
oX
3
þ
o
u
3
oX
1
ou
1
oX
1
ou
1
oX
3
þ
o
u
2
oX
1
ou
2
oX
3
þ
o
u
3
oX
1
ou
3
oX
3
ð
3
:
73
Þ
þ
1
2
:
G
23
¼
G
32
¼
1
2
ou
2
oX
3
þ
o
u
3
oX
2
ou
1
oX
2
ou
1
oX
3
þ
o
u
2
oX
2
ou
2
oX
3
þ
o
u
3
oX
2
ou
3
oX
3
Analogue to tensor representation (
3.69
), it can be seen using (
3.72
) and (
3.73
)
that the elements of the principal diagonal G
ii
of G are dependent on the squares of
the displacement derivations, i.e.
ð
ou
1
=
oX
1
Þ
2
and so on, and the elements of the
secondary diagonal G
ij
are dependent on the bilinear terms of the displacement
derivations, i.e.
ð
ou
1
=
oX
1
Þð
ou
1
=
oX
2
Þ
and so on.
Rigid Body Motion. Under the motion of a rigid body, the displacement vectors
of all material points are equal due to u
ð
X
;
t
Þ¼
u
0
. They are no longer position
dependent (cf. Fig.
3.8
b). According to (
3.56
), the displacement gradient vanishes
due to H
¼
u
0
r¼
0
:
Using (
3.57
) this yields the deformation gradient to be
F
¼
I
:
ð
3
:
74
Þ
Using (
3.74
) together with (
3.65
) and (
3.69
), the strain tensors yield
C
¼
B
¼
I
and
G
¼
0
ð
3
:
75
Þ
where, in the case of no deformation (rigid body motion), only the GREEN-strain
tensor transforms into the zero tensor, and thus represents a ''true'' strain measure.
Furthermore, according to (
3.70
), (
3.53
) and (
3.55
) it remains dx
¼
dX and
dV
¼
dV
0
, which leads to constant edge lengths of a volume element and the
volume element itself remains unchanged in case of a rigid body motion. The body
thus does not experience any deformation.
