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 R1 and R2.

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).