Biomedical Engineering Reference
In-Depth Information
location of a crack initiation and the direction of its propagation are dependent on
the inherent strength of the material and on the magnitude of the imposed stresses
within the material. It has been suggested that for a bone subjected to pure torsional
loading, failure begins with a crack initiation in a shear mode, followed by a crack
propagation generally along the line of a maximum tensile stress perpendicular to
the neutral axis.
Consider a cylindrical object
(Figure 5.14) subject to equal and opposite torque
(
T
) at both ends. To illustrate the effect of torsion, a square is drawn the surface.
With the application of torsion on the cylinder, the circular symmetry of the object
is maintained, which implies in turn that plane cross-sections remain plane, without
warping. However, the square element shows shear-type deformation twisted by an
angle
. The shear stress varies from zero in the neutral axis to a maximum at the
outside surface of the cylindrical structure. On the contrary, if a diamond-shaped
schematic is drawn on the surface of the cylinder, then the diamond-shaped element
experiences a deformation analogous to simple tension and compression; it elon-
gates and narrows, with maximum tensile stresses acting on a plane perpendicular
to the axis of elongation and maximum compressive stresses orthogonal to that.
Performing a momentum balance similar to bending moment derivation, the
shear stress (
θ
τ
) due to torsion is given by
Tr
J
τ =
(5.37)
where
J
is the polar moment of inertia of the cross-section. Similar to the area mo-
ment of inertia, the polar moment of inertia is a measure of resistance offered by an
object to torsion. The polar momentum of inertia is defined by
∫
2
J
=
r A
Neutral axis
Neutral axis
T
Shear
Max tensile stress
Max shear stress
Compression
Max comp stress
Tension
T
Unloaded
Loaded in torsion
(a)
(b)
Figure 5.14
Effect of torsional loading cylindrical material: (a) before application and (b) after ap-
plication and stress distribution.
