**
12.11 SURFACE FATIGUE
**

All
the surface-failure modes discussed above apply to situations in which the
relative motions between the surfaces are essentially pure sliding, such as a
cam running against a flat-faced follower. When two surfaces are in
**
pure rolling
**
contact, or are primarily rolling in combination with a small
percentage of sliding, a different surface failure mechanism comes into play,
called
**
surface fatigue
**
. Many applications of this condition exist
such as cams with roller followers, ball and roller bearings, nip rolls, and
spur or helical gear tooth contact. All except the gear teeth and nip rolls
typically have essentially pure rolling with only about 1% sliding.

The
stresses introduced in two materials contacting at a rolling interface are
highly dependent on the geometry of the surfaces in contact as well as on the
loading and material properties. The general case allows any three-dimensional
geometry on each contacting member and, as would be expected, its calculation
is the most complex. Two special-geometry cases are of practical interest and
are also somewhat simpler to analyze. These are
*
sphere-on-sphere
*
and
*
cylinder-on-cylinder
*
. In all cases, the radii of curvature of the
mating surfaces will be significant factors. By varying the radii of curvature
of one mating surface, these special cases can be extended to include the
sub-cases of
*
sphere-on-plane
*
,
*
sphere-in-cup,
cylinder-on-plane
*
, and
*
cylinder-in-trough
*
. It is only necessary to make the radii of
curvature of one element infinite to obtain a plane, and negative radii of
curvature define a concave cup, or concave trough surface. For example, some
ball bearings can be modeled as
*
sphere-on-plane
*
and some roller bearings and
cylindrical cam followers as
*
cylinder-in-trough
*
.

As
a ball passes over another surface, the theoretical contact patch is a point of
zero dimension. A roller against a cylindrical or flat surface theoretically
contacts along a line of zero width. Since the area of each of these
theoretical contact geometries is zero, any applied force will then create an
infinite stress. We know that this cannot be true, as the materials would
instantly fail. In fact, the materials must deflect to create sufficient
contact area to support the load at some finite stress. This deflection creates
a semi-ellipsoidal pressure distribution over the contact patch. In the general
case, the contact patch is elliptical as shown in Figure 12-11
*
a
*
.
Spheres will have a circular contact patch, and cylinders create a rectangular
contact patch as shown in Figure 12-11
*
b
*
.

Consider
the case of a spherical ball rolling in a straight line against a flat surface
with no slip, and under a constant normal load. If the load is such as to
stress the material only below its yield point, the deflection in the contact
patch will be elastic and the surface will return to its original curved
geometry after passing through contact. The same spot on the ball will contact
the surface again on each succeeding revolution. The resulting stresses in the
contact patch are called
**
contact
stresses or Hertzian stresses
**
.
The contact stresses in this small volume of the ball are
**
repeated
**
at the ball’s rotation frequency. This creates a fatigue-loading
situation that eventually leads to a
**
surfacefatigue
failure
**
.

This
repeated loading is similar to a tensile fatigue-loading case. The significant
difference in this case is that the principal contact stresses at the center of
the contact patch are all compressive, not tensile. Fatigue failures are
considered to be initiated by shear stress and continued to failure by tensile
stress. There is also a shear stress associated with these compressive contact
stresses, and it is believed to be the cause of crack formation after many
stress-cycles. Crack growth then eventually results in failure by
**
pitting
**
—
*
the fracture and dislodgment of small pieces
of material from the surface
*
.
Once the surface begins to pit, its surface finish is compromised and it
rapidly proceeds to failure by
**
spalling
**
—
*
the
loss of large pieces of the surface
*
.
Figure 12-12 shows some examples of pitted and spalled surfaces.

If
the load is large enough to raise the contact stress above the material’s
compressive yield strength, then the contact-patch deflection will create a
permanent flat on the ball. This condition is sometimes called
**
false brinelling
**
, because it has a similar appearance to the indentation
made to test a material’s Brinell hardness. Such a flat on even one of its
balls (or rollers) makes a ball (or roller) bearing useless.

We
will now investigate the contact-patch geometries, pressure distributions,
stresses, and deformations in rolling contacts starting with the relatively
simple geometry of
*
sphere-on-sphere
*
, next dealing with the
*
cylinder-on-cylinder
*
case, and finally discussing the
*
general
*
case. Derivation of the equations for these cases are among the
more complex sets of examples from the theory of elasticity. The equations for
the area of contact, deformation, pressure distribution, and contact stress on
the centerline of two bodies with static loading were originally derived by
Hertz in 1881,[9] an English translation of which can be found in [10]. Many
others have since added to the understanding of this
problem.[11],[12],[13],[14]

Copyright 2004, Industrial
Press, Inc., New York, NY