Page 38 - Tribology in Machine Design

Basic principles of tribology 25

where d is the distance between the centres of the two hemispheres in
contact and x denotes the position of the moving hemisphere. By
substitution of eqn (2.22) into eqns (2.20) and (2.21), the load, P, and the
area of contact, A, may be estimated at any time.
Denoting by a the angle of inclination of the load P on the contact with
the horizontal, it is easy to find that

The total horizontal and vertical forces, H and V, at any position defined by
x of the sliding asperity (moving linearly past the stationary one), are given
by

Equation (2.24) can be solved for different values of d and /?.
A limiting value of the geometrical interference w can be estimated for the
initiation of plastic flow. According to the Hertz theory, the maximum
contact pressure occurs at the centre of the contact spot and is given by

The maximum shear stress occurs inside the material at a depth of
approximately half the radius of the contact area and is equal to about
0.31go- From the Tresca yield criterion, the maximum shear stress for the
initiation of plastic deformation is Y/2, where Y is the tensile yield stress of
the material under consideration. Thus

Substituting P and A from eqns (2.20) and (2.21) gives

Since Y is approximately equal to one third of the hardness for most
materials, we have

where (f) = RiR 2/(Ri + #2) an d Hb denotes Brinell hardness.
The foregoing equation gives the value of geometrical interference, w, for
the initiation of plastic flow. For a fully plastic junction or a noticeable
plastic flow, w will be rather greater than the value given by the previous
relation. Thus the criterion for a fully plastic junction can be given in terms