Page 39 - Tribology in Machine Design

26 Tribology in machine design

of the maximum geometric interference

Hence, for the junction to be completely plastic, w max must be greater than
vv p. An approximate solution for normal and shear stresses for the plastic
contacts can be determined through slip-line theory, where the material is
assumed to be rigid-plastic and nonstrain hardening. For hemispherical
asperities, the plane-strain assumption is not, strictly speaking, valid.
However, in order to make the analysis feasible, the Green's plane-strain
solution for two wedge-shaped asperities in contact is usually used. Plastic
deformation is allowed in the softer material, and the equivalent junction
angle a is determined by geometry. Quasi-static sliding is assumed and the
solution proposed by Green is used at any time of the junction life. The
stresses, normal and tangential to the interface, are

where a is the equivalent junction angle and y is the slip-line angle.
Assuming that the contact spot is circular with radius a, even though the
Green's solution is strictly valid for the plane strain, we get

where a = x/2(/>w and (t> — RiR 2/(Ri + R2)- Resolution of forces in two fixed
directions gives

where <5 is the inclination of the interface to the sliding velocity direction.
Thus V and H may be determined as a function of the position of the
moving asperity if all the necessary angles are determined by geometry.

2.10 The probability of As stated earlier, the degree of separation of the contacting surfaces can be
surface asperity contact measured by the ratio h/cr, frequently called the lambda ratio, L In this
section the probability of asperity contact for a given lubricant film of
thickness h is examined. The starting point is the knowledge of asperity
height distributions. It has been shown that most machined surfaces have
nearly Gaussian distribution, which is quite important because it makes the
mathematical characterization of the surfaces much more tenable.
Thus if x is the variable of the height distribution of the surface contour,
shown in Fig. 2.11, then it may be assumed that the function F(x), for the
cumulative probability that the random variable x will not exceed the