Archard equation

The Archard equation is a simple model used to describe sliding wear and is based around the theory of asperity contact.

Equation

:$Q\; =\; frac\; \{KW\}H$

where:: "Q" is the total volume of wear debris produced per unit distance moved: "W" is the total normal load: "H" is the hardness: "K" is a dimensionless constant

Derivation

The equation can be derived by first examining the behavior of a single asperity.:The local load $,\; delta\; W$, supported by an asperity, assumed to have a circular cross-section with a radius $,\; a$, is:

:$delta\; W\; =\; P\; pi\; \{a^2\}\; ,!$

where "P" is the yield pressure for the asperity, assumed to be deforming plastically. "P" will be close to the indentation hardness, "H", of the asperity.

If the volume of wear debris, $,\; delta\; V$, for a particular asperity is a hemisphere sheared off from the asperity, it follows that:: $delta\; V\; =\; frac\; 2\; 3\; pi\; a^3$

This fragment is formed by the material having slid a distance 2"a"

Hence, $,\; delta\; Q$, the wear volume of material produced from this asperity per unit distance moved is:

:$delta\; Q\; =\; frac\; \{delta\; V\}\; \{2a\}\; =\; frac\; \{pi\; a^2\}\; 3\; equiv\; frac\; \{delta\; W\}\; \{3P\}\; approx\; frac\; \{delta\; W\}\; \{3H\}$ making the approximation that $,P\; approx\; H$

However, not all asperities will have had material removed when sliding distance 2"a". Therefore, the total wear debris produced per unit distance moved, $,\; Q$ will be lower than the ratio of "W" to "3H". This is accounted for by the addition of a dimensionless constant "K", which also incorporates the factor 3 above. These operations produce the Archard equation as given above.

"K" is therefore a measure of the severity of wear. Typically for 'mild' wear, "K" ≈ 10⁻⁸, whereas for 'severe' wear, "K" ≈ 10⁻².