Page 73 - Analysis and Design of Machine Elements

Figure 2.8 Contact stresses on contacting Strength of Machine Elements 51
cylinders.

loads. The magnitude of contact stress depends on the material properties, body geome-
tries and applied loads [18]. Generally, small radii lead to small contact areas and, con-
sequently, large contact stress.
Figure 2.8 illustrates the contact area and corresponding contact stress distribution
within two cylinders with diameters of and , loaded with uniformly distributed
1 2
force F along cylinder length b. The contact area is a narrow rectangle. The induced
contact stress is three-dimensional but may be characterized by stress perpendicular to
the plane of contact surface, with parabolic distribution, varying from the periphery of
contact surface to the maximum at the centre [1]. Assume the contact is frictionless and
the contacting bodies are elastic, isotropic and homogeneous. The maximum contact
stress is calculated by the Hertz formula as [14]
√
( )
√ 1 1
√ ±
√ F 1 2
√
Hmax = √ • ( ) (2.51)
b 1− 2 1− 2
1 + 2
E 1 E 2
where , E and are Poisson’s ratio, elastic modulus and radius of curvature, respec-
tively. The subscripts 1 and 2 refer to the two cylinders. The negative symbol ‘–’ refers
to an internal surface. The Hertz formula is the foundation for the calculation of load
carrying capabilities of rolling contact bearings, gears and so on.
2.4.2 Surface Fatigue Failure

In machine elements like rolling contact bearings or mating gear teeth, contact stresses
produced by repeated application of loads are cyclic in nature. The cyclic contact
stresses, together with material defects, lead to subsurface minute cracks initially.
During each cycle of operation, lubricants are forced into the minute cracks under
pressure, causing cracks propagate over time and eventually resulting in small bits
of material dislodged from the surface, leaving cracks, pits or flaking on the element
surfaces, as illustrated in Figure 2.9. This surface damage is called surface fatigue failure.