Archard equation derivation
For asperity contact, the local load δW, supported by an asperity, assumed to be circular in cross-section with a radius a, is
δW = P π a2
where P is the yield pressure for the asperity, assumed to be deforming plastically. As assumed when developing a theory of friction, P will be close to the indentation hardness, H, of the asperity (which will be an asperity on the softer surface).
As sliding proceeds, wear will arise from the continuous formation and destruction of asperity contacts.
If, for a particular asperity, the volume of wear debris, δV, is a hemisphere sheared off from the asperity, it follows that
δV = 2/3π a3
This fragment is formed by the material having slid a distance 2a.
Hence, δQ, the wear volume of material produced from this asperity / unit distance moved is simply
\[\delta Q = \frac{{\delta V}}{{2a}} = \frac{{\pi {a^2}}}{3} = \frac{{\delta W}}{{3P}} \approx \frac{{\delta W}}{{3H}}\]
making the approximation that P ≈ H.
However, not all asperities will have had material removed in a sliding operation. The total volume of wear debris produced per unit distance moved, Q, will therefore be lower than the ratio of the total normal load, W, to 3H. It is convenient to write this dimensionless constant of proportionality as a constant K with the factor 3 subsumed into K, giving the so-called Archard equation:
\[Q = \frac{{KW}}{H}\]