P. Pallav, PhD

Research into the nature of physical contact dates back to about 1500 AD, when the great scientist Leonardo da Vinci was doing his experiments on friction. He found that the force (F_f) required to cause sliding is proportional to the force (F_n) perpendicular to the contact. This is expressed as

The parameter f is called a constant, because it is independent of the normal force, but does depend on the materials and circumstances.

Leonardo could not explain this and it would take until 1950, when Bowden and Tabor published their contact hypothesis, which has now been accepted widely.
To explain Equation 1, they introduced the concept of Actual Area of Contact and Apparent Area of Contact.

Apparent vs. Actual Contact Area

According to the contact hypothesis, normal contacting surfaces have an arbitrary roughness. When two surfaces approach, the first contact will appear where two roughness asperities touch.

Because of the small dimensions of such a micro contact, any small force will generate very high stresses. The micro contact must yield in a way that allows the bodies to come closer. The pressure on such a plastically deformed micro contact is about equal to the surface hardness value of the softest surface. Therefore in this way more microscopic contacts must arise until the total area of contact is large enough to support the normal force, so

Consequently, what (macroscopically) appears to be a contact point or line, is in fact a set of a very large number of microscopic contacts.
The total amount of actual contact area is usually substantially smaller than the apparent area of contact and will approach it only under extreme circumstances (like making an indentation to perform a hardness measurement).

Example:

A 10-millimeter gold alloy cube (VHN ≈ 500 MPa), laying on a plane composite surface with a weight of 0.02 kg (≈ 0.2 N), has an apparent area of contact of 100 mm² and a total actual contact area of only 400 µm², which is 250,000 (!) times as small. In the example in the page on surface fatigue of a gold alloy cylinder and a plane surface of composite, where the normal force is 1 N, the actual contact area amounts a substantial (!) 5 % (≈2000 µm²) of the 1 mm x 40 µm (=40,000 µm²) 'Hertz' apparent contact area.

Friction

At the Actual Area of Contact the materials are pressed together at a usually rather high pressure. In the example above this pressure is about 5000 times the atmospheric pressure! At these circumstances some bonding (adhesion) will occur and these bonds or junctions will have a certain shear strength, which will result in friction:

Combining Equations 1 through 3 we get

And Bowden and Tabor said QED. The frictional force depends only on the normal force, and the ratio between these mainly depends on two system parameters; the hardness of the softest surface and the shear bond strength of a cold pressed junction between the substrate materials.

Bowden FP and Tabor D (1950): The Friction and Lubrication of Solids. Clarendon Press, Oxford.