**
12.12 SPHERICAL CONTACT
**

Cross
sections of two spheres in contact are shown in Figure 12-13. The dotted lines
indicate the possibilities that one is a flat plane or a concave cup. The
difference is only in the magnitude or the sign of its radius of curvature
(convex +, concave –). Figure 12-11
*
a
*
(p. 353) shows the general
semi-ellipsoidal pressure distribution over the contact patch. For a
sphere-on-sphere, it will be a hemisphere with a circular contact patch (
*
a
*
=
*
b
*
).

**
FIGURE 12-12
**

Examples of surfaces
failed by pitting and spalling due to surface fatigue (Source: J. D. Graham,
Pitting of Gear Teeth, in Handbook of Mechanical Wear, C. Lipson, ed., U. Mich.
Press, 1961, pp. 138, 143, with permission)

**
Contact Pressure and Contact Patch in Spherical
Contact
**

The
contact pressure is a maximum
*
pmax
*
at the center and zero at the
edge. The total applied load
*
F
*
on the contact patch is equal
to the volume of the hemisphere:

*
*

where
*
a
*
is the half-width (radius) of the contact
patch. This can be solved for the maximum pressure:

*
*

The
average pressure on the contact patch is the applied force divided by its area:

*
*

and
substituting equation 12.8c in 12.8b gives:

*
*

We
now define
*
material constants
*
for the two spheres

*
*

where
*
E
*
1,
*
E
*
2 and
1,
2 are the Young’s moduli and Poisson’s ratios
for the materials of sphere 1 and sphere 2, respectively.

The
dimensions of the contact area are typically very small compared to the radii
of curvature of the bodies, which allows the radii to be considered constant
over the contact area despite the small deformations occurring there. We can
define a
*
geometry constant
*
that depends only on the radii
*
R
*
1
and
*
R
*
2 of the two spheres,

*
*

To
account for the case of a sphere-on-plane,
*
R
*
2 becomes
infinite, making 1/
*
R
*
2 zero. For a sphere-in-cup,
*
R
*
2
becomes negative. (See Figure 12-13, p. 355.) Otherwise
*
R
*
2
is finite and positive, as is
*
R
*
1.

The
contact-patch radius
*
a
*
is then found from

*
*

Substitute
equation 12.8b in 12.9c:

*
*

The
pressure distribution within the hemisphere is

*
*

We
can normalize the pressure
*
p
*
to the magnitude of
*
pavg
*
and
the patch dimension
*
x
*
or
*
y
*
to the patch radius
*
a
*
and
then plot the normalized pressure distribution across the patch, which will be
an ellipse as shown in Figure 12-14.

**
**

**
Static Stress Distributions in Spherical Contact
**

The
pressure on the contact patch creates a three-dimensional stress state in the
material. The three applied stresses
*
x
*
,
*
y
*
,
and
*
z
*
are
compressive and are maximal at the sphere’s surface in the center of the patch.
They diminish rapidly and nonlinearly with depth and with distance from the
axis of contact. They are called
**
Hertzian
stresses
**
in honor of their
original discoverer. A complete derivation of these equations can be found in
reference 16. Note that these applied stresses in the
*
x, y
*
,
and
*
z
*
directions are also the principal stresses in
this case. If we look at these stresses as they vary along the
*
z
*
axis
(with
*
z
*
increasing into the material) we find

Pressure distribution
across contact patch

Poisson’s
ratio is taken for the sphere of interest in this calculation. These normal
(and principal) stresses are maximal at the surface, where
*
z
*
=
0:

There
is also a principal shear stress induced from these principal normal stresses:

which
is not maximum at the surface but rather at a small distance
*
z
*
@
ô
*
max
*
below the surface.

Figure
12-15 shows a plot of the principal normal and maximum shear stresses as a
function of depth
*
z
*
along a radius of the sphere. The stresses are
normalized to the maximum pressure
*
pmax
*
, and the depth is normalized to the half-width
*
a
*
of the contact patch. This plot provides a
dimensionless picture of the stress distribution on the centerline under a
spherical contact. Note that all the stresses have diminished to less than10%
of
*
pmax
*
within
*
z
*
= 5
*
a
*
.
The subsurface location of the maximum shear stress can also be seen. If both
materials are steel, it occurs at a depth of about 0.63
*
a
*
and
its magnitude is about 0.34
*
pmax
*
. The shear stress is about 0.11
*
pmax
*
at
the surface on the
*
z
*
axis.

The
subsurface location of the maximum shear stress is believed by some to be a
significant factor in surface-fatigue failure. The theory says cracks that
begin below the surface eventually grow to the point that the material above
the crack breaks out to form a pit as shown in Figure 12-12 (p. 354).

Normalized stress
distribution along z axis in spherical contact; xyz stresses are principal

*
*

Figure
12-16 shows a photoelastic model of the contact stresses in a cam immediately
beneath a loaded roller follower.[15] Experimental photoelastic stress analysis
uses a physical model of the part to be analyzed made from a transparent
plastic material (Lexan in this example) that shows fringes of constant stress
magnitude when loaded and viewed in polarized light. The maximum shear stress
can be clearly seen a small distance into the cam directly under the follower.
While this is a cylindrical rather than a spherical contact, their stress
distributions along the centerline are similar, as will be seen in the next
section.

When
we move off the centerline of the contact patch on the surface of the sphere,
the stresses diminish. At the edge of the patch the radial stress
*
z
*
is
zero, but there is a condition of pure shear stress with the magnitude:

The
two nonzero principal stresses will be
*
xy
*
,
which means that there is also a tensile

stress
at that point of

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