Biomedical Engineering Reference
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will be different from the length of d
X
, in general. To calculate this change in length
we denote the square of the final length by d
s
2
¼
d
x
d
x
and the square of the initial
length by d
S
2
¼
d
X
d
X
. The difference in the squares of these length changes is
written d
s
2
d
S
2
can be written
entirely in terms of the material filament d
X
or entirely in terms of its image d
x
at
time
t
. To accomplish both of these objectives we observe that, using d
x
d
S
2
d
X
. This expression for d
s
2
¼
d
x
d
x
d
X
¼
F
d
X
,
d
x
d
x
has the representation
F
T
dx
dx
¼ð
F
dX
Þð
F
dX
Þ¼
dX
F
dX
;
(11.12)
F
1
and that d
X
¼
d
x
,d
X
d
X
has the representation
F
1
T
F
1
F
1
F
1
dx
dX
dX
¼ð
dx
Þð
dx
Þ¼
dx
:
(11.13)
The two formulas may then be derived from the expression d
s
2
d
S
2
¼
d
x
d
x
d
X
d
X
, one by substituting for d
x
d
x
from (
11.12
) and the other by substituting
for d
X
d
X
from (
11.13
), thus
ds
2
dS
2
F
T
¼
dx
dx
dX
dX
¼f
F
1
g
dX
dX
F
1
T
F
1
¼f
1
g
dx
dx
:
(11.14)
The Lagrangian or material strain tensor
E
and the Eulerian or spatial strain
tensor
e
are defined by
fF
T
f1 ðF
1
1
2
1
2
T
F
1
E ¼
F 1g;
e ¼
Þ
g:
(11.15)
In terms of the strain tensors the change d
s
2
d
S
2
takes the form
ds
2
dS
2
¼
2
dX E
dX ¼
2
dx e
dx:
(11.16)
Clearly, if either of these strain tensors is zero, then so is the other and there is no
change in length for any material filament, d
s
2
d
S
2
.
The Lagrangian strain tensor
E
and Eulerian strain tensor
e
are defined for large
strains and represented in terms of the deformation gradient and its inverse in
(
11.15
). A component representation of these two tensors in terms of the displace-
ment vector
u
will now be obtained. From (2.20) and Fig. 2.4,
u
is given by
u
¼
c
.If
u
is referred to the spatial coordinate system and the spatial
gradient taken, or if
u
is referred to the material coordinate system and the material
gradient taken (Sect. 2.2), then
½r
o
¼
x
X
þ
T
T
F
1
u
ð
X
;
t
Þ
¼
F
ð
X
;
t
Þ
1
and
½r
u
ð
x
;
t
Þ
¼
1
ð
x
;
t
Þ:
(2.23)
repeated
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