Biomedical Engineering Reference
In-Depth Information
continuum is given by:
x
1
¼
X
I
þ
X
II
t
þ
X
III
t
2
;
2.2.2. The motion of
a
X
I
t
2
X
II
t
2
x
2
¼
X
II
þ
X
III
t
þ
;
x
3
¼
X
III
þ
X
I
t
þ
:
(a) Find the inversion of this motion.
(b) Determine
the velocity and the
acceleration in the material
representation.
(c) Find the velocity in the spatial (Eulerian) representation for this motion.
(d) Find the three tensors
L
,
D
, and
W
for this motion.
(e) Find the tensor of deformation gradients
F
for this motion.
2.2.3. The motion of a continuum is given by:
x
1
¼
X
I
þ
X
II
sin
ðp
t
Þ;
x
2
¼
X
II
X
I
sin
ðp
t
Þ;
x
3
¼
X
III
:
(a) Determine the deformation gradient
F
of this deformation.
(b) Determine the instantaneous configuration image of the set of points
(
X
I
)
2
(
X
II
)
2
1 in the reference configuration.
(c) Describe the geometry of the set of points (
X
I
)
2
þ
¼
(
X
II
)
2
1 in the
reference configuration and describe what happens to this set of points
in the motion of the continuum as time
t
increases.
þ
¼
2.3
Infinitesimal Motions
The term infinitesimal motion is used to describe the case when the deformation,
including rotation, is small. This does not mean that the displacement vector is
small; one can have large displacements but small strain infinitesimal motions.
Large displacements associated with small strain infinitesimal motions occur in
very thin long rods. The criterion for infinitesimal motion is that the square of the
gradients of displacement be small compared to the gradients of displacement
themselves. Thus, for infinitesimal motions, the squares and products of the nine
quantities
@
u
1
x
1
;
@
u
1
x
2
;
@
u
1
x
3
;
@
u
2
x
1
;
@
u
2
x
2
;
@
u
2
x
3
;
@
u
3
x
1
;
@
u
3
x
2
;
@
u
3
(2.37)
@
@
@
@
@
@
@
@
@
x
3
2
must be small compared to their own values. This means, for example,
f@
u
2
=@
x
1
g
is required to be much smaller than
x
1
; each such square and product of these
nine quantities is so small that it may be neglected compared to the quantity itself.
Using this criterion of smallness, representations of the kinematics variables for
infinitesimal motions will be developed in this section.
If the motion is infinitesimal the deformation gradient tensor
F
must not deviate
significantly from the unit tensor
1
, the magnitude of the deviation being restricted
by the criterion on the deformation gradients stated in the previous paragraph.
@
u
2
=@
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