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
1*a *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