Archard equation

Equation

Q = K W L H

where:

Q is the total volume of wear debris produced K is a dimensionless constant W is the total normal load L is the sliding distance H is the hardness of the softest contacting surfaces

Note that W L is proportional to the work done by the friction forces as described by Reye's hypothesis.

Derivation

The equation can be derived by first examining the behavior of a single asperity.

The local load δ W , supported by an asperity, assumed to have a circular cross-section with a radius a , is:

δ W = P π 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, δ V , for a particular asperity is a hemisphere sheared off from the asperity, it follows that:

δ V = 2 3 π a 3

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

Hence, δ Q , the wear volume of material produced from this asperity per unit distance moved is:

δ Q = δ V 2 a = π a 2 3 ≡ δ W 3 P ≈ δ W 3 H making the approximation that P ≈ H

However, not all asperities will have had material removed when sliding distance 2a. 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.Archard interpreted K factor as a probability of forming wear debris from asperity encounters. Typically for 'mild' wear, K ≈ 10⁻⁸, whereas for 'severe' wear, K ≈ 10⁻². Recently, it has been shown that there exists a critical length scale that controls the wear debris formation at the asperity level. This length scale defines a critical junction size, where bigger junctions produce debris, while smaller ones deform plastically.