Biomedical Engineering Reference
In-Depth Information
C
T
¼ð
F
T
F
Þ
T
¼
F
T
F
T
T
¼
F
T
F
¼
C
B
T
¼ð
F
F
T
Þ
T
¼
F
T
T
F
T
¼
F
F
T
¼
B
ð
3
:
65
Þ
Remarks: Following (
3.53
), (
3.63
) may be interpreted as C being the square of
the line element dX (from the ICFG), mapped into the square of dx in the CCFG.
For B
1
the reverse applies. The difference of (
3.65
) and (
3.45
) is that the former
represent
H
T
H
H
H
T
,
nonlinear
strain
measures
due
to
the
terms
and
respectively.
Taking (
3.59
) and (
3.60
) into account, the following relations between C and
U and B and V, respectively, yield:
C
¼
F
T
F
¼ð
R
U
Þ
T
ð
R
U
Þ¼ð
U
T
R
T
Þð
R
U
Þ¼
U
R
T
R
U
¼
U
U
¼
U
2
|{z}
I
B
¼
F
F
T
¼ð
V
R
Þð
V
R
Þ
T
¼ð
V
R
Þð
R
T
V
T
Þ¼
V
R
R
T
V
¼
V
V
¼
V
2
|
{z
}
I
ð
3
:
66
Þ
As applicable to the one dimensional case, change in length is used to quantify
strain. One kind of length change arises employing (
3.63
)
1
and (
3.64
)
1
by calcu-
lating the difference of the squares of the line elements dx and dX by (note for
arbitrary vectors the identity v
¼
v
I holds)
ð
dx
Þ
2
ð
dX
Þ
2
¼
dX
C
dX
dX
dX
¼
dX
C
dX
dX
I
dX
¼
dX
ð
C
I
Þ
|{z}
2G
ð
3
:
67
Þ
dX
and thus
ð
dx
Þ
2
ð
dX
Þ
2
¼
dX
2G
dX
ð
3
:
68
Þ
where considering (
3.64
)
1
, the abbreviation (G is symmetric!)
G :
¼
1
2
ð
C
I
Þ¼
1
2
ð
H
þ
H
T
þ
H
T
H
Þ
1
2
½
u
rþr
u
þðr
u
Þð
u
rÞ¼
G
T
ð
3
:
69
Þ
is referred to as the right G
REEN
or G
REEN
-L
ANGRANGE
-strain tensor. Due to the
symmetry of C and I, the right G
REEN
-strain tensor is also symmetric. To clarify
the composition of the right G
REEN
-strain tensor, in the following G is represented
with respect to an OBS following (
3.69
): analogue to (
3.51
), one obtains obeying
(
3.58
)
þ
1
2
e
i
e
j
G
¼
G
ij
e
i
e
j
¼
1
2
ou
i
oX
j
þ
ou
j
oX
i
o
u
k
oX
i
o
u
k
oX
j
ð
3
:
70
Þ
