The independent variables in these equations are then the coordinates x, z in the cross section of the roller, referenced to the contact point, the half-width a of the contact patch, and the maximum normal load pmax at  the contact point.

Equations 12.22 (pp. 369–370) define the behavior of the stress functions below the surface, but when z = 0,  the factors  become infinite and these equations fail. Other forms are needed to account for the  tresses on the surface of the contact patch.

The total stress on each Cartesian plane is found by superposing the components due to the normal and tangential loads:

For short rollers in plane stress, y is zero, but if the rollers are long axially, then a plane strain condition will exist away from the ends and the stress in the y direction will be:

where  is Poisson’s ratio.

These stresses are maximum at the surface and decrease with depth. Except at very low ratios of tangential force to normal force (< about 1/9)[14],[17] the maximum shear stress occurs at the surface as well, unlike the pure rolling case. A computer program was written to evaluate equations 12.22 and 12.23 (pp. 369–370) for the conditions at the surface and plot them. The stresses are all normalized to the maximum normal load pmax and the locations normalized to the patch half-width a . A coefficient of friction of 0.33 and steel rollers with í = 0.28 were assumed for the examples. The magnitudes and shapes of the stress distributions will be a function of these factors.

Figure 12-19 a shows the x direction stresses at the surface, which are due to the normal and tangential loads, and also shows their sum from the first of equations 12.24a. Note that the stress component xt due to the tangential force, is tensile from the contact point to and beyond the trailing edge of the contact patch. This should not be surprising, as one can picture that the tangential force is attempting to pile up material in front of the contact point and stretch it behind that point, just as a carpet bunches up in front of anything you try to slide across it. The stress component xn due to the normal force is compressive everywhere. However, the sum of the two x components has a significant normalized tensile value of twice the coefficient of friction (here 0.66 pmax ) and a compressive peak of about –1.2 pmax . Figure 12-19 b shows all of the applied stresses in x, y , and z directions across the surface of the contact zone. Note that the stress fields on the surface extend beyond the contact zone when a tangential force is present, unlike the situation in pure rolling where they extend beyond the contact zone only under the surface. (See Figure 12-17 on p. 361)

Figure 12-20 shows the principal and maximum shear stresses for the plane strain, applied stress state in Figure 12-19. Note that the magnitude of the largest compressive principal stress is about 1.38 pmax and the largest tensile principal stress is 0.66 pmax at the trailing edge of the contact patch. The presence of an applied tangential shear stress in this example increases the peak compressive stress by 40% over a pure rolling case and introduces a tensile stress in the material. The principal shear stress reaches a peak value of 0.40 pmax at x / a = 0.4. All the stresses shown in Figures 12-19 and 12-20 are at the surface of the rollers.

Beneath the surface, the magnitudes of the compressive stresses due to the normal load reduce. However, the shear stress xzn due to the normal loading increases with depth, becoming a maximum beneath the surface at z = 0.5 a, as shown in Figure 12-21. Note the sign reversal at the midpoint of the contact zone. There are fully reversed shear stress components acting on each differential element of material as it passes through the contact zone. The peak-to-peak range of this fully reversed shear stress in the xz plane is greater in magnitude than the range of the maximum shear stress and is considered by some to be responsible for subsurface pitting failures.[13]

FIGURE 12-19

Applied tangential, normal, and shear stresses at surface for cylinders in combined rolling and sliding with   = 0.33

Figure 12-22 plots the principal and maximum shear stresses (calculated for  = 0.33 and a plane strain condition) versus the normalized depth z / a taken at the x / a = 0.3 plane (where the principal stresses are maximum as shown in Figure 12-20). All the stresses are maximum at the surface. The principal stresses diminish rapidly with depth, but the shear stress remains nearly constant over the first 1a of depth.

At the surface, the maximum shear stress is relatively uniform across the patch width with a peak of 0.4 at x / a = 0.4 when  = 0.33, as shown in Figure 12-20. This max peak location moves versus the patch centerline with increasing depth, but its magnitude varies only slightly with depth.

FIGURE 12-20

Principal stresses across contact zone at surface for cylinders in combined rolling and sliding with ì = 0.33

FIGURE 12-21

Shear stresses below surface at z / a = 0.5 for cylinders in combined rolling and sliding with   = 0.33