Archard equation

The Archard wear equation is a simple model used to describe sliding wear and is based on the theory of asperity contact. The Archard equation was developed much later than Reye's hypothesis [it] (sometimes also known as energy dissipative 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. Theodor Reye's model^[1][2] became popular in Europe and it is still taught in university courses of applied mechanics.^[3] Until recently, Reye's theory of 1860 has, however, been totally ignored in English and American literature^[3] where subsequent works by Ragnar Holm^[4][5][6] and John Frederick Archard are usually cited.^[7] In 1960, Mikhail Mikhailovich Khrushchov [ru] and Mikhail Alekseevich Babichev published a similar model as well.^[8] In modern literature, the relation is therefore also known as Reye–Archard–Khrushchov wear law. In 2022, the steady-state Archard wear equation was extended into the running-in regime using the bearing ratio curve representing the initial surface topography.^[9]

Equation

${\displaystyle Q={\frac {KWL}{H}}}$ 

where:^[10]

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 ${\displaystyle WL}$  is proportional to the work done by the friction forces as described by Reye's hypothesis.

Also, K is obtained from experimental results and depends on several parameters. Among them are surface quality, chemical affinity between the material of two surfaces, surface hardness process, heat transfer between two surfaces and others.

Derivation

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

The local load ${\displaystyle \,\delta W}$ , supported by an asperity, assumed to have a circular cross-section with a radius ${\displaystyle \,a}$ , is:^[11]

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

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, ${\displaystyle \,\delta V}$ , for a particular asperity is a hemisphere sheared off from the asperity, it follows that:

${\displaystyle \delta V={\frac {2}{3}}\pi a^{3}}$ 

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

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

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

However, not all asperities will have had material removed when sliding distance 2a. Therefore, the total wear debris produced per unit distance moved, ${\displaystyle \,Q}$  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.^[12] Typically for 'mild' wear, K ≈ 10⁻⁸, whereas for 'severe' wear, K ≈ 10⁻². Recently,^[13] 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.

See also

• Chemistry of pressure-sensitive adhesives – Chemical science associated with pressure-sensitive adhesives

References

1. Reye, Karl Theodor (1860) [1859-11-08]. Bornemann, K. R. (ed.). "Zur Theorie der Zapfenreibung" [On the theory of pivot friction]. Der Civilingenieur - Zeitschrift für das Ingenieurwesen. Neue Folge (NF) (in German). 6: 235–255. Retrieved 2018-05-25. [1]
2. Rühlmann, Moritz (1979) [1885]. Manegold, Karl-Heinz; Treue, Wilhelm (eds.). Vorträge über Geschichte der Technischen Mechanik und Theoretischen Maschinenlehre sowie der damit im Zusammenhang stehenden mathematischen Wissenschaften, Teil 1. Reihe I. - Darstellungen zur Technikgeschichte (in German) (reprint of 1885 ed.). Hildesheim / New York: Georg Olms Verlag (originally by Baumgärtner's Buchhandlung, Leipzig). p. 535. ISBN 978-3-48741119-4. Retrieved 2018-05-20. {{cite book}}: |work= ignored (help) (NB. According to this source Theodor Reye was a polytechnician in Zürich in 1860, but later became a professor in Straßburg.)
3. Villaggio, Piero [in Italian] (May 2001). "Wear of an Elastic Block". Meccanica. 36 (3): 243–249. doi:10.1023/A:1013986416527. S2CID 117619127. [2]
4. Holm, Ragnar (1946). Electrical Contacts. Stockholm: H. Gerber.
5. Holm, Ragnar; Holm, Else (1958). Electric Contacts Handbook (3rd completely rewritten ed.). Berlin / Göttingen / Heidelberg, Germany: Springer-Verlag. ISBN 978-3-66223790-8. [3] (NB. A rewrite and translation of the earlier "Die technische Physik der elektrischen Kontakte" (1941) in German language, which is available as reprint under ISBN 978-3-662-42222-9.)
6. Holm, Ragnar; Holm, Else (2013-06-29) [1967]. Williamson, J. B. P. (ed.). Electric Contacts: Theory and Application (reprint of 4th revised ed.). Springer Science & Business Media. ISBN 978-3-540-03875-7. (NB. A rewrite of the earlier "Electric Contacts Handbook".)
7. Ponter, Alan R. S. (2013-09-09). "Re: Is wear law really Archard's law (1953), or Reye's law (1860)?". from the original on 2018-05-28. Retrieved 2018-05-28. Jack was a Reader at Leicester until he retired in the early 1980s and ran a successful experimental tribology research program. He was very meticulous and I very much doubt if he had heard of Reye's work, particularly as it wasn't published in English. It is quite common for ideas to appear independently in different countries over time.
8. Хрущов [Khrushchov], Михаил Михайлович [Mikhail Mikhailovich] [in Russian]; Бабичев [Babichev], Михаил Алексейевич [Mikhail Alekseevich] (1960), Issledovaniya iznashivaniya metallov Исследования изнашивания металлов [Investigation of wear of metals] (in Russian), Moscow: Izd-vo AN SSSR (Russian academy of sciences)
9. Varenberg, Michael (2022). "Adjusting for Running-in: Extension of the Archard Wear Equation". Tribology Letters. 70 (2): 59. doi:10.1007/s11249-022-01602-6. S2CID 248508580.
10. Archard, John Frederick (1953). "Contact and Rubbing of Flat Surface". Journal of Applied Physics. 24 (8): 981–988. Bibcode:1953JAP....24..981A. doi:10.1063/1.1721448.
11. "DoITPoMS - TLP Library Tribology - the friction and wear of materials. - Archard equation derivation". www.doitpoms.ac.uk. Retrieved 2020-06-14.
12. Archard, John Frederick; Hirst, Wallace (1956-08-02). "The Wear of Metals under Unlubricated Conditions". Proceedings of the Royal Society. A-236 (1206): 397–410. Bibcode:1956RSPSA.236..397A. doi:10.1098/rspa.1956.0144. S2CID 135672142.
13. Aghababaei, Ramin; Warner, Derek H.; Molinari, Jean-Francois (2016-06-06). "Critical length scale controls adhesive wear mechanisms". Nature Communications. 7: 11816. Bibcode:2016NatCo...711816A. doi:10.1038/ncomms11816. PMC 4897754. PMID 27264270.

Further reading

• Peterson, Marshall B.; Winer, Ward O. (1980). Wear Control Handbook. New York: American Society of Mechanical Engineers (ASME).
• Friction, Lubrication, and Wear Technology. ASM Handbook. 1992. ISBN 978-0-87170-380-4.
• Panetti, Modesto [in Italian] (1954) [1947]. Meccanica Applicata (in Italian). Torino: Levrotto & Bella.
• Funaioli, Ettore (1973). Corso di meccanica applicata alle macchine (in Italian). Vol. I (3rd ed.). Bologna: Patron.
• Funaioli, Ettore; Maggiore, Alberto; Meneghetti, Umberto (October 2006) [2005]. Lezioni di meccanica applicata alle macchine (in Italian). Vol. I. Bologna: Patron. ISBN 978-8-85552829-0.
• Ferraresi, Carlo; Raparelli, Terenziano (1997). Meccanica Applicata (in Italian) (C.L.U.T. ed.). Torino.{{cite book}}: CS1 maint: location missing publisher (link)
• Opatowski, Izaak [in Esperanto] (September 1942). "A theory of brakes, an example of a theoretical study of wear". Journal of the Franklin Institute. 234 (3): 239–249. doi:10.1016/S0016-0032(42)91082-2.
• https://patents.google.com/patent/DE102005060024A1/de (Mentions the term "Reye-Hypothese")