# Classical friction

In 1699, Amontons rediscovered Leonardo da Vinci's two laws of friction:
the frictional force is directly proportional to the normal load, and the
size of the bodies does not affect the friction [Bowden and Tabor, 1950; 1974].
Desaguliers introduced the idea that adhesion also plays a part, which was
later verified by Coulomb. The surface of any solid, no matter how
polished, has many asperities sticking out (Figure 3.1). It
is these asperities that make contact with the other surface: putting
two solids together is rather like turning Switzerland upside down and
standing it on Austria - the area of intimate contact will be small.''
[Bowden and Tabor, 1974]

• Figure 3.1: Contact of two surfaces,
like turning Switzerland upside down and
standing it on Austria''.

When two metal surfaces are brought together the area of
asperity-to-asperity contact is extremely small so the pressure is very
high. Even at small contact loads plastic deformation occurs at the
asperities, while the underlying metal still deforms elastically. As the
normal load is increased the asperities deform and fracture, thus
increasing the real area of contact, the sum of all the surface
irregularities that touch and support the load. This is much less than
the apparent area which remains unchanged. Use of the usual assumption
that the local plastic yield pressure is constant, gives the
real area of contact for one asperity under a load to be
, so the total real contact area is:

where N is the total normal force. Hence, for a metal, the real area of
contact is proportional to the load and independent of the size of the
surfaces.

To shear these two bodies apart by sliding, a force F=SA is needed,
where S is the shear breaking strength of the contacts. Hence
and hence Amontons' law.

Assuming a von Mises yield surface () gives
. Alternatively, assuming a Tresca yield surface

()
gives . (From a Mohr's circle for simple shear,
and
.)
Experimental data on metals fall between the two
surfaces, but in general are closer to the von Mises yield surface
[e.g. Lu, 1996].
Assuming pure metal on metal contact and a von Mises yield surface would

Most metal surfaces are covered by a thin film of oxide, water vapour and
other absorbed impurities. Where the asperities contact, the metal
surfaces weld together to form junctions. The shear strength of these
junctions is heavily dependent on the shear strength of the surface
films. Thus it is the surface oxide layer that determines the coefficient of
friction and not, in general, the parent metal [Ashby and Jones, 1980].

If the actual area of contact is increased, for example by heating the
metals in a vacuum [Bowden and Tabor, 1974] (or by applying a large electric field for
nylon on glass [Bradbury and Reicher, 1952]), then the coefficient of friction can be
increased by an order of magnitude. Table 3.1 from
Ashby and Jones [1980] lists ranges of for various materials.

 Material Perfectly clean metals in vacuum seizure Clean metals in air 0.8-2 Clean metals in wet air 0.5-1.5 Steel on dry bearing materials 0.1-0.5 (e.g. lead, bronze) Steel on ceramics 0.1-0.5 (e.g. sapphire, diamond, ice) Ceramics on ceramics 0.05-0.5 (e.g. carbides on carbides) Polymers on polymers 0.05-1.0 Metals and ceramics on polymers 0.04-0.5 Boundary lubrication of metals 0.05-0.2 Hydrodynamic lubrication 0.001-0.005