Thomas Industrial Library

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