Models of friction
Simple model of friction for metals
To account for the observed values of μ, we need to take account of the stress system at the contact areas, and consider how this changes as the frictional force is applied.
In a simple model of metal-metal contact, we can imagine that at each asperity contact there is an area of contact, a, that acts like a hardness indentation, with an indentation pressure, P, equal to the hardness, H, of the asperity material. The normal force on the contact, w, is then w = a H.
Summing over all asperity contacts between two surfaces, the total normal force, W = A H, where A is the true area of contact (not the nominal area of contact). We can assume that, to a good approximation, H = 3σy, the uniaxial yield stress.
Using the Tresca criterion, the yield stress in pure shear, k, is half the uniaxial yield stress, σy.
Hence, when sliding is just about to occur, the total shear force, F, is such that
\[F = \frac{{{\sigma _{\rm{y}}}}}{2}A = \frac{{{\sigma _{\rm{y}}}}}{2}\frac{W}{H} = \frac{{{\sigma _{\rm{y}}}}}{2}\frac{W}{{3{\sigma _{\rm{y}}}}} = \frac{W}{6}\]
and so since W is a normal force, we predict that
\[\mu = \frac{1}{6}\]
This simple model gives the correct order of magnitude for a coefficient of friction between two metal surfaces. It also shows that the frictional force is independent of the nominal area and proportional to the load, W, because of the way in which the true area of contact varies as a function of nominal area and applied load.
Measured values are often greater than this. This is due to work hardening (flow stress of the material/asperity will not be constant, it will increase) and/or junction growth (the area of the contact increasing).
If there were a thin interfacial contaminant layer with shear yield stress τi at the contact regions, then a straightforward modification of the above analysis would predict that
\[\mu = \frac{{{\tau _{\rm{i}}}}}{{3{\sigma _{\rm{y}}}}}\]
Note that this model is not relevant to either polymers or ceramics. For polymers, asperities are soft and elastic, so at high contact pressures the true area of contact can approach the nominal area of contact. For ceramics, fracture can arise at the asperity tops, and so the plastic flow model is not appropriate.
A more sophisticated theory of friction
When there is no frictional force, the stress state, σ, can be assumed to be of the form
\[\sigma = \left( {\begin{array}{*{20}{c}}{{p_0}}&0&0\\0&0&0\\0&0&0\end{array}} \right)\]
for a normal stress p0. With a frictional force, the stress state can be assumed to take the form
\[\sigma = \left( {\begin{array}{*{20}{c}}{{p_1}}&\tau &0\\\tau &0&0\\0&0&0\end{array}} \right)\]
To obtain plastic flow in this situation, the area of contact, A, increases, to recognise the plastic flow of the asperity arising from the compressive stress imposed on it.
For a yield criterion to be satisfied and plastic flow to occur, p1 = W/A and τ = F/A.
Using the Tresca yield criterion we find
\[\mu = \frac{{{F_{\max }}}}{W} = \frac{1}{{2{{\left( {{{\left( {{\tau _0}/{\tau _{\rm{i}}}} \right)}^2} - 1} \right)}^{{\rm{ }}1/2}}}}\]
where τ0 = shear strength of bulk material and τi = shear strength of weak interfacial film at interface.
If τ0 >> τi, this model predicts
\[\mu \approx \frac{{{\tau _{\rm{i}}}}}{{2{\tau _0}}} = \frac{{{\tau _{\rm{i}}}}}{{{\sigma _y}}}\]