Thomas Industrial Library

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]