**
12.14 GENERAL CONTACT
**

When
the geometry of the two contacting bodies is allowed to have any general
curvature, the contact patch is an ellipse and the pressure distribution is a
semi-ellipsoid, as shown in Figure 12-11
*
a
*
(p. 353). Even
the most general curvature can be represented as a radius of curvature over a
small angle with minimal error. The size of the contact patch for most
practical materials in these applications is so small that this approximation
is reasonable. Thus the compound curvature of each body is represented by two
mutually orthogonal radii of curvature at the contact point.

**
**

**
Contact Pressure and Contact Patch in General
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 semi-ellipsoid:

*
*

where
*
a
*
is the half-width of the major axis and
*
b
*
the
half-width of the minor axis of the contact-patch ellipse. 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 12.18c in 12.18b gives

*
*

We
must define three geometry constants that depend on the radii of curvature of
the two bodies,

where
*
R
*
1 and
*
R
*
1’ are the two radii of curvature* of body 1,
*
R
*
2
and
*
R
*
2
*
’
*
are the radii* of body 2, and
is the angle between the planes containing *R*1
and *R*2.

The
contact-patch dimensions
*
a
*
and
*
b
*
are then found from

* Measured in mutually perpendicular planes

where
*
m
*
1 and
*
m
*
2 are material constants as defined in equation
12.9a (p. 355) and the values of
*
ka
*
and
*
kb
*
are
from Table 12-2 corresponding to the value of
from equation 12.19c.

The
pressure distribution within the semi-ellipsoid is

*
*

which
is an ellipse as shown in Figure 12-11 (p. 353).

**
**