Biomedical Engineering Reference
In-Depth Information
The polar moments of inertia for common cross-sections are summarized in
Table 5.5. Using Hooke's law, the shear stress is related to the shear strain (i.e., the
rotation of the cross sections). The corresponding shear strain (
γ
) is given by
Tr
GJ
γ =
(5.38)
The product
GJ
is called the
torsional rigidity
. Typically, the failure plane of
a brittle shaft under torsion is often at a 45° angle with respect to the shaft's axis.
EXAMPLE 5.9
A skier with a mass of 65 kg catches the edge of the right ski while making a turn. The
torque developed during the accident is 15 Nm. The minimum cross-section of the tibia is
approximated by a tube of elliptical shape with an outer major axis of 20 mm and a minor
axis of 16 mm. The inner major axis is 18 mm and the minor axis is 14.5 mm. If the aver-
age ultimate shear strength is 60 MPa, determine whether the tibia will fracture or not.
Solution:
T
= 1 Nm
c
= 0.01m (outer radius of the major axis)
J
= polar moment of inertia of the elliptical tube =
J
out
-
J
in
Using the relation for
J
from Table 5.5 for elliptical geometry,
π
π
(
)
(
)
2
2
2
2
4
12
4
J
* 10 * 8 10
8
* 9 * 7.25 9
7.25
3,460 mm
3,460 * 10
−
m
=
+
−
+
=
=
4
4
Tc
J
15 * 0.01
43.35 MPa
Using (5.37),
τ
==
=
xy
3, 460 * 10
−
12
Since it is below the ultimate shear strength, the tibia would not break.
5.4
Nonidealities in Stress-Strain Characterization
5.4.1 Failure Under Combined Loading
Tension, compression, shear, bending, and torsion represent simple and pure modes
of loading. In reality, all loading patterns are observed in the body when activities
are performed. Injuries encountered in the body are more commonly a product of
a combination of the aforementioned modes exposed and short-term and long-
term loadings and movements combined. Furthermore, the mode of loading is de-
termined by the direction of the load application and the fact that in the case of
traumatic injuries (e.g., automobile) there is virtually no constraint on applied load
orientation (or magnitude). In addition, all materials show anisotropic mechanical
behavior owing to their anatomical adaptations. For example, bone exhibits greater
strength when subjected to tension directed longitudinally versus tension directed
transversely. Under tensile loading, the orientation of the fracture line is perpendic-
ular to the long axis of the bone. Unlike failure in tension, a compressive failure in
