Biomedical Engineering Reference
In-Depth Information
application position (radial and axial) within the material. Assuming that the elastic limit
(in shear) of the material is not reached and that the applied moment is constant, the mate-
rial will deform to a certain extent (this would be seen as a twist of the bar). With removal
of the moment, the solid material will return to its original conformation. Recall that the
shear stresses that are generated in a torsional bar are similar to what we will learn devel-
ops in a fluid; however, the mechanisms of loading conditions and stress development are
fundamentally different. We use this example because it should be familiar from previous
courses and should highlight that some assumptions, such as geometry, may be relaxed in
physiological environments.
In contrast to the analysis just described for solid materials, a fluid subjected to a shear-
ing force will deform continually. A typical example (from a fluid dynamics course) to
depict a fluid under a shear loading condition is with a parallel plate design with a thin
layer of fluid between the plates (
Figure 2.2
). Imagine that a force is applied to the top
plate, which causes it to move with some velocity in the positive X direction, to one partic-
ular fixed location. The fluid that is in contact with the top plate will move with the same
velocity as the plate because of the no-slip boundary condition. (Later we will see that the
no-slip boundary condition can lead to a zero velocity along the wall.) No-slip dictates
that a fluid layer in contact with a solid boundary must have the same velocity as that
boundary. Therefore, because of this applied force, a fluid element denoted by ABCD will
deform to ABC
0
D
0
. However, if one looks at a later time, the fluid element will continue to
deform, even though the applied force is no longer acting and the top plate is no longer
moving as a result of the applied force F. The fluid element can then be depicted at the
later time as the element ABC
. In biology, a condition depicted in
Figure 2.2
can occur
in the lubrication of joints, where one solid boundary (a bone) moves in relation to another
solid boundary (a second bone). One of these boundaries could be considered stationary
through the use of a translating coordinate system fixed to that boundary. (This simplifies
our analysis instead of having two moving boundaries.) The other boundary will then
move with some velocity relative to the stationary boundary. This discussion highlights
one of the fundamental differences between solid and fluid materials: solids will deform
to one particular conformation under a constant shearing force, whereas fluids continually
deform under a constant shear force. Recall that there are viscoelastic materials that exhibit
properties of solids and fluids both and, therefore, continually deform under constant
loading to some threshold deformation state.
Fluid mechanics can be divided into two main categories: static fluid mechanics and
dynamic fluid mechanics. In static fluid mechanics, the fluid is either at rest or is undergo-
ing rigid-body motion. Therefore, the fluid elements are in the same arrangement at all
times. The fluid element also does not experience any type of deformation (linear or
v
D
v
FIGURE 2.2
F
Under a constant shearing force (F), a fluid
will continually deform.
Immediately after the application of
a constant force to the top plate, a fluid element ABCD, deforms
to ABC
0
D
0
. At a later time, the element will have deformed to
ABC
CC΄ C˝
D D΄ D˝
even though the force F is held constant and no addi-
tional force has been added to the system.
v
D
v
A
B
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