Herttzian Mechanics (contd...)

Let's say two solids are in point contact initially, which on application of load now contacts through an area. Before deformation the separation between two corresponding points and is given by equation 8.5.

Due to the symmetry of this expression about O, the contact region spans equal distance on its either side. During compression two distant points and move towards each other along the axis by distances and respectively.

Furthermore, due to the contact pressure the surfaces of the two bodies are displaced by distances and respectively. Then considering that the two points and coincide at the surface, we should have,

	(8.8)

Writing and making use of 8.5,

	(8.9)

If and lie outside the contact area, so that they do not touch each other,

	(8.10)

Putting and , equation 5.8 can be written in dimensionless form:

	(5.11)

Putting and writing , the deformation within the contact area becomes,

	(5.12)

Given that the deformation is small, i.e. , the state of strain in each solid is characterized by the ratio . Now the magnitude of the strain will be proportional to the contact pressure divided by the elastic modulus. If is the average pressure acting mutually on each solid, equation 5.11 becomes,

i.e.		(8.13)