Archard equation
Encyclopedia
The Archard wear equation is a simple model
used to describe sliding wear
and is based around the theory of asperity contact. The Archard equation was developed later than the Reye's hypothesis, though both came to the same physical conclusions, that the volume of the removed debris due to wear is proportional to the work done by friction forces. Reye’s model became very popular in Europe and it is still taught in university courses of applied mechanics. This theory has, however, been totally ignored in English and American literature where subsequent works by Ragnar Holm and John F. Archard are usually cited.
where:
Note that
is proportional to the work done by the friction forces as described by Reye's hypothesis.
The local load
, supported by an asperity, assumed to have a circular cross-section with a radius 
, is:
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,
, for a particular asperity is a hemisphere sheared off from the asperity, it follows that:
This fragment is formed by the material having slid a distance 2a
Hence,
, the wear volume of material produced from this asperity per unit distance moved is:
making the approximation that 
However, not all asperities will have had material removed when sliding distance 2a. Therefore, the total wear debris produced per unit distance moved,
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.
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
used to describe sliding wear
Wear
In materials science, wear is erosion or sideways displacement of material from its "derivative" and original position on a solid surface performed by the action of another surface....
and is based around the theory of asperity contact. The Archard equation was developed later than the Reye's hypothesis, though both came to the same physical conclusions, that the volume of the removed debris due to wear is proportional to the work done by friction forces. Reye’s model became very popular in Europe and it is still taught in university courses of applied mechanics. This theory has, however, been totally ignored in English and American literature where subsequent works by Ragnar Holm and John F. Archard are usually cited.
Equation
where:
- Q is the total volume of wear debris produced
- W is the total normal load
- H is the hardness of the softest contacting surfaces
- K is a dimensionless constant
- L is the sliding distance
Note that
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
, supported by an asperity, assumed to have a circular cross-section with a radius , is: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,
, for a particular asperity is a hemisphere sheared off from the asperity, it follows that:
This fragment is formed by the material having slid a distance 2a
Hence,
, the wear volume of material produced from this asperity per unit distance moved is: making the approximation that However, not all asperities will have had material removed when sliding distance 2a. Therefore, the total wear debris produced per unit distance moved,
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.