Page 97 - Tribology in Machine Design

Elements of contact mechanics 83

2
first becomes a maximum. Mathematically, this is where d fi/dTl is the first
3
maximum value or the minimum value of /?, where d /?/dT;? =0. Thus,
starting with eqn (2.67) it is possible to derive the following expression
for T c

Equation (3.20) is implicit and must be solved by using a microcomputer,
for instance, in order to obtain values for T c.

3.7.6. The case of circular contact
Archard has presented a simple formulation for the mean flash temperature
in a circular area of real contact of diameter 2a. The friction energy is
assumed to be uniformly distributed over the contact as shown schemati-
cally in Fig. 3.7. Body 1 is assumed stationary, relative to the conjunction
area and body 2 moves relative to it at a velocity V. Body 1, therefore,
receives heat from a stationary source and body 2 from a moving heat
source. If both surfaces move (as with gear teeth for instance), relative to the
conjunction region, the theory for the moving heat source is applied to both
bodies.
Archard's simplified formulation also assumes that the contacting
portion of the surface has a height approximately equal to its radius, a, at
the contact area and that the bulk temperature of the body is the
temperature at the distance, a, from the surface. In other words, the
contacting area is at the end of a cylinder with a length-to-diameter ratio of
approximately one-half, where one end of the cylinder is the rubbing surface
and the other is maintained at the bulk temperature of the body. Hence the
model will cease to be valid, or should be modified, as the length-to-
diameter ratio of the slider deviates substantially from one-half, and/or as
the temperature at the root of the slider increases above the bulk
temperature of the system as the result of frictional heating. If these
assumptions are kept in mind, Archard's simplified formulation can be of
value in estimating surface flash temperature, or as a guide to calculations
with modified contact geometries.
For the stationary heat source, body 1, the mean temperature increase
Figure 3.7
above the bulk solid temperature is

l
where Q i is the rate of frictional heat supplied to body 1, (Nm s ), k l is the
thermal conductivity of body 1, (W/m °C) and a is the radius of the circular
contact area, (m).
If body 2 is moving very slowly, it can also be treated as essentially a