Biomedical Engineering Reference
In-Depth Information
compare the action of solids and fluids under a shear stress loading condition, the distinc-
tion is clear: a solid will deform under shear, but this deformation does not continue when
the shearing force is removed or held constant. However, as a caveat to that statement,
some materials exhibit solid and fluid properties both. These materials are termed “visco-
elastic” materials, and they deform continually under shear loading until some threshold
deformation has been reached. In fact, all real solid materials must exhibit some fluid
properties, but the fluid-like properties can be neglected in many situations.
We will begin our discussion with a review of shear loading, which should be familiar
from a solid mechanics course. A typical manner to induce a shear loading condition on a
solid material is through torsion. Torsion is defined as the twisting of a material when it is
loaded by moments (or torques) that produce a rotation about an axis through the mate-
rial. In solid mechanics courses, torsion analysis is typically applied to a solid bar fixed on
one end and a moment (M) applied at some location along the length of the bar
( Figure 2.1 , which illustrates the loading at the free end of the bar). We will not define the
appropriate equations of state for this loading condition. However, please refer to a text-
book on solid mechanics for an appropriate review. (We suggest some textbooks in the
Further Readings section.) The same analysis that is applied to a fixed solid bar can be
applied to a bone subjected to torsion. Imagine an athlete who plants his or her foot on a
surface while making a quick turn about that foot. This would generate a moment
throughout the bones in the leg. To solve for the shear forces/stresses within the bone,
assume that the foot is fixed to the surface and that the bone is modeled as a hollow cylin-
der (with a taper, if necessary, to make the solution more accurate) ( Figure 2.1 , which illus-
trates different modeling methods used in biomedical engineering). This simplifying
assumption ignores inhomogeneities (if any) in the bone material properties. We will see
that this assumption is made in many biofluid mechanics examples because it is very diffi-
cult to mathematically quantify inhomogeneities within a fluid (we typically assume a uni-
form distribution of molecules throughout the fluid). In cases of pure torsion, the shear
forces are dependent on the applied moment, the geometry of the material and the load
FIGURE 2.1
A depiction of a method to
model torsion throughout a solid material.
Through the application of a moment at a location
on the bar, a line AB would deform to the arc
AB 0 . With removal of the moment, point B 0 would
move back to B, assuming the induced stress did
not exceed the materials yield stress. The second
figure depicts a cylinder, which may be used for
torsion analysis in bones.
M
Lo ngitudinal
axis
A
B
M
Lo ngitudinal
axis
A
B
 
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