Page 289 - Tribology in Machine Design

274 Tribology in machine design

and as such is unlikely to be an important factor in the failure of the
material. The maximum shear stress occurs at a small depth inside the
material, and has a value of 0.3p max; at the surface, the maximum shear
stress is 0.25p max. However, when sliding is introduced, a tangential stress
field due to friction is added to the normal load. As the friction increases, the
region of maximum shear stress (located at half the contact area radius
beneath the surface), moves upwards whilst simultaneously a second region
of high yield stress develops on the surface behind the circle of contact. The
shear stress at the surface is sufficient to cause flow when the coefficient of
friction reaches about 0.27. These stresses are much more likely to be
responsible for the failure of the gear teeth. The important point for the
designer at this stage is that each of these stresses is proportional to p max,
and therefore for any given material are proportional to -JP/R.
For several reasons, however, this result cannot be directly applied to
gear teeth. The analysis assumes two surfaces of constant radii of curvature,
and an elastic homogeneous isotropic stress-free material. First, a ^,car tooth
profile has a continuously varying radius of curvature, and the importance
of this departure from the assumption may be emphasized by considering
the case of an involute tooth where the profile starts at the base circle. The
radius of curvature, say R lf is at all times the length of the generating
tangent, so at this point it is, from a mathematical point of view, zero; but it
remains zero for no finite length of the involute curve, growing rapidly as we
go up the tooth and having an unknown value within the base circle. If
contact were to occur at this point the stress would not be infinite, as an
infinitely small distortion would cause the load to be shared by the
adjoining part of the involute profile, so that there would be a finite area of
contact. Clearly, the Hertz analysis is rather inapplicable at this point; all
that can be said is that the stresses are likely to be extremely high. In the
regions where contact between well-designed gear teeth does occur the rate
of change of R^ is much less rapid, and it is not unreasonable to take a mean
value at any instant for the short length in which we are interested.
Second, the assumption that the material is elastic will certainly break
down if the resulting shear stress exceeds the shear yield strength of the
material. The consequences are quite beyond our ability to predict them
mathematically. We might manage the calculations if one load application
at one instant were all we had to deal with; but the microscopic plastic flow
which then occurred would completely upset our calculations for contact at
the next point on the tooth profile and so on. The situation when the
original contact recurred would be quite different; and we have to deal with
millions of load cycles as the gears revolve. All that can be said is that the
repeated plastic flow is likely to lead to fatigue failure, but that it will not
necessarily do so, since the material may perhaps build up a favourable
system of residual stress, and will probably work-harden to some extent. If
such a process does go on then there is no longer a homogeneous isotropic
stress-free material.
Third, gears which are transmitting more than a nominal power must be
lubricated. The introduction of a lubricating film between the surfaces