Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Tribology - the friction and wear of materials Estimate for the true area of contact
Tribology - the friction and wear of materials AimsBefore you startIntroductionSurface topographyFriction - recapFriction - properties of the coefficient of friction, μFriction theoryLubrication - introduction and types of lubricantsLubrication - additivesWear - introductionWear by hard particles - abrasion and erosionSummaryQuestionsGoing further
TLP creditsTLP contentsShow all content
Viewing and downloading resourcesAbout the TLPsTerms of useFeedbackCredits Print this page
PreviousNext

Estimate for the true area of contact

Consider the deformation of a single spherical asperity, of radius r, pressed against a plane surface under a load w:

Fig

The radius of contact is a. The area of contact, πa2, is A. For purely elastic deformation under conditions of plane strain

\[A = \pi {a^2} = \pi {\rm{ }}{\left( {\frac{{3wr}}{{4E}}} \right)^{2/3}}\] where \[\frac{1}{E} = \frac{{1 - \nu _{{\rm{sphere}}}^2}}{{{E_{{\rm{sphere}}}}}} + \frac{{1 - \nu _{{\rm{surface}}}^2}}{{{E_{{\rm{surface}}}}}}\] and r is the asperity radius

Note that for either the sphere or the material it is indenting, \(E/(1 - {\nu ^2})\) is the Young’s modulus in plane strain.

For perfect plastic deformation (e.g., when the asperity has yielded),

\[A \propto w\]

which is the same as in hardness testing.

Multiple asperity contact:

Extending the principles found in single asperity contacts to multiple asperities requires a statistical theory of multiple asperity contact by two rough surfaces.

It has been found that

\[A \propto {w^{1 - \delta }}\]

with δ << 1, even for elastic contact. Hence, to a very good approximation, the ratio w/A is almost constant.

From models it has also been able to deduce whether asperities make contact elastically or plastically. The asperity deformation mode is found to be dependent on the value of a plasticity factor, y:

\[\psi  = \frac{E}{H}{\left( {\frac{{\sigma *}}{r}} \right)^{1/2}}\]

where E is the Young’s modulus, H is hardness, σ* is the standard deviation of the distribution of asperity heights and r is the asperity radius (assumed in this model for y to be the same for all the asperities).

\(\frac{{\sigma *}}{r}\) \(\approx {\rm{ }}\)mean asperity slope

The dependence of asperity deformation mode with \(\psi \) for aluminium with different surface roughness values is shown below.

Fig

For:

\(\psi  \le {\rm{ 0}}{\rm{.6}}\) the behaviour is almost perfectly totally elastic

\(\psi  \ge {\rm{ 1}}\) the behaviour is almost perfectly totally plastic

It will only be for very fine polishing (small \(\frac{{\sigma *}}{r}{\rm{ }}\)) that asperity contact between aluminium surfaces will remain elastic. For ‘average’ metallurgical polishes plastic deformation will arise at asperity contacts.

However, for polymers and ceramics,

\[\frac{E}{H} \approx {\rm{ 0}}{\rm{.1 }}{\left( {\frac{E}{H}} \right)_{{\rm{metals}}}}\]

and this leads to a greater likelihood of elastic contact between asperities. This is why wear of ceramics tends to be dominated by brittle fracture of surface material.

The important result from this is that for most surfaces, i.e., for a given σ*, r and surface density of asperities, the deformation mode (elastic or plastic) cannot be altered by changes in the load.

PreviousNext