Biomedical Engineering Reference
In-Depth Information
respectively. Using the chain rule for partial derivatives it is easy to verify that
F
1
is indeed the inverse of
F
,
FF
1
F
1
F
¼
¼
1
:
(2.17)
Recall that any motion can be decomposed into a sum of a translational,
rotational, and deformational motion. The deformation gradient tensors remove
the translational motion as may be easily seen because the translational motion is a
separate function of time (cf., e.g.,
2.2
) that must be independent of the particle
X
.
Thus only the rotational motion and the deformational motion determine
F
.If
F
¼
1
there are no rotational or deformational motions. If
F
¼
Q
(
t
),
Q
(
t
)
Q
(
t
)
T
1
, it follows from (
2.4
) that the motion is purely rotational and there is no
deformational motion. The
deformation gradient F
is so named because it is a
measure of the deformational motion as long as
F
¼
6¼
Q
(
t
). If
F
¼
Q
(
t
), then the
motion is rotational and we replace
F
by
Q
(
t
).
The determinant of the tensor of deformation gradients, J, is the Jacobian of the
transformation from
x
to
X
, thus
Det
F
1
J
Det
F
¼
=
;
1
(2.18)
where it is required that
0
<
J
<1
(2.19)
so that a finite continuum volume always remains a finite continuum volume.
If
c
represents the position vector of the origin of the coordinate system used for
the configuration at time
t
relative to the origin of the coordinate system used for the
configuration at
t
¼
0, then the displacement vector
u
of the particle
X
is given by
(Fig.
2.2
),
u
¼
x
X
þ
c
:
(2.20)
The displacement vectors
u
for all the particles
X
O
(0) are given by
u
ð
X
;
t
Þ¼wð
X
;
t
Þ
X
þ
c
ð
t
Þ;
X
O
ð
0
Þ;
(2.21)
or by
w
1
u
ð
x
;
t
Þ¼
x
ð
x
;
t
Þþ
c
ð
t
Þ;
x
O
ð
t
Þ:
(2.22)
Two gradients of the displacement field
u
may then be calculated, one with
respect to the spatial coordinate system
x
denoted by the usual gradient symbol
∇
and one with respect to the material coordinate system
X
denoted by the gradient
symbol
∇
O
, thus
T
T
F
1
½r
O
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)
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