Biomedical Engineering Reference
In-Depth Information
are no longer functions of time so they are denoted by
A
,
B
,
C
,
D
,
E
, and
F
. They are
evaluated by substituting the initial and final locations of the set of particles,
X
ð
1
Þ
I
ð
,
X
ð
1
Þ
II
X
ð
2
Þ
I
,
X
ð
2
Þ
II
X
ð
3
Þ
I
,
X
ð
3
Þ
II
x
ð
1
1
,
x
ð
1
Þ
x
ð
2
1
,
Þ¼
(0, 0),
ð
Þ¼
(1, 0),
ð
Þ¼
(0, 1), and
ð
Þ¼
(1, 2),
ð
2
x
ð
2
2
Þ¼
x
ð
3
Þ
1
,
x
ð
3
2
Þ¼
(2.5, 3.75), respectively, into the last set of equations
in Example 2.1.2. The values obtained are
A
(2, 3.25),
ð
¼
1,
B
¼
1.75,
C
¼
1.5,
D
¼
1.25,
E
2 and they are obtained by substituting the values for the relevant
points given in the statement of the problem above into the last set of equations in
Example 2.1.2. The planar homogeneous deformation then has the representation
¼
1, and
F
¼
x
1
¼
2
X
I
þ
1
:
5
X
II
þ
1
;
x
2
¼
1
:
25
X
I
þ
1
:
75
X
II
þ
2
;
x
3
¼
X
III
;
which is a particular case of (
2.10
) To double check this calculation one can check
to see if each marker is mapped correctly from its initial position to its final
position.
There are two coordinate systems with respect to which a gradient may be taken,
either the spatial coordinate system
x
,(
x
1
,
x
2
,
x
3
), or the reference material coordi-
nate system
X
,(
X
I
,
X
II
,
X
III
). To distinguish between gradients with respect to these
two systems, the usual gradient symbol
will be used to indicate a gradient with
respect to the spatial coordinate system
x
, and the gradient symbol
∇
∇
O
with a
subscripted boldface
O
will indicate a gradient with respect to the material coordi-
nate system
X
. The (material) deformation gradient tensor
F
is defined by
T
F
¼½r
O
wð
X
;
t
Þ
for all
X
O
ð
0
Þ:
(2.13)
The (spatial) inverse deformation gradient tensor
F
1
is defined by
T
F
1
¼½rw
1
ð
x
;
t
Þ
for all
x
O
ð
t
Þ;
(2.14)
where
X ¼ w
1
ðx;
t
Þ
for all
x
O
ð
t
Þ
(2.15)
is the inverse of the motion (
2.2
). The components of
F
and
F
1
are
2
3
2
3
@
x
1
@
x
1
@
x
1
@
X
I
@
X
I
@
X
I
4
5
4
5
@
X
I
@
X
II
@
X
III
@
x
1
@
x
2
@
x
3
¼
¼
@
x
i
@
x
2
@
x
2
@
x
2
@
X
a
@
@
X
II
@
@
X
II
@
X
II
@
and
F
1
F
¼
¼
@
X
a
x
i
@
X
I
@
X
II
@
X
III
x
1
@
x
2
x
3
@
x
3
@
x
3
@
x
3
@
X
III
@
@
X
III
@
@
X
III
@
@
X
I
@
X
II
@
X
III
x
1
x
2
x
3
(2.16)
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