Page 105 - Tribology in Machine Design

Elements of contact mechanics 91

application of a load, it is seen that the normal approach will be given by
(z — d), where d is the current separation between the smooth surface and
the reference plane. Clearly, each asperity is deformed equally and carries
the same load W t so that for rj asperities per unit area the total load W will
be equal to rjW t. For each asperity, the load W t and the area of contact A (
are known from the Hertz theory

and

where d is the normal approach and R is the radius of the sphere in contact
with the plane. Thus if /? is the asperity radius, then

and the total load will be given by

that is the load is related to the total real area of contact, A=riA t, by

This result indicates that the real area of contact is related to the two-thirds
power of the load, when the deformation is elastic.
If the load is such that the asperities are deformed plastically under a
constant flow pressure H, which is closely related to the hardness, it is
assumed that the displaced material moves vertically down and does not
spread horizontally so that the area of contact A' will be equal to the
geometrical area 2n^d. The individual load, W' t, will be given by

that is, the real area of contact is linearly related to the load.
It must be pointed out at this stage that the contact of rough surfaces
should be expected to give a linear relationship between the real area of
contact and the load, a result which is basic to the laws of friction. From the
simple model of rough surface contact, presented here, it is seen that while a
plastic mode of asperity deformation gives this linear relationship, the
elastic mode does not. This is primarily due to an oversimplified and hence