Most models of surface contact consider the surface roughness to be on one of the contacting surfaces only. The authors give a general theory of contact between two rough plane surfaces. They show that the important results of the previous models are unaffected: in particular, the load and the area of contact remain almost proportional, independently of the detailed mechanical and geometrical properties of the asperities. Further, a single-rough-surface model can always be found which will predict the same laws as a given two-rough-surface model, although the required model may be unrealistic. It does not seem possible to deduce the asperity shape or deformation mode from the load-compliance curve.

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Bradley (1932) showed that if two rigid spheres of radii R 1 and R 2 are placed in contact, they will adhere with a force 2πR∆γ, where R is the equivalent radius R 1 R 2 /(R 1 +R 2 ) and ∆γ is the surface energy or 'work of adhesion' (equal to γ 1 +γ 2 − γ 12 ). Subsequently Johnson et al. (1971) (JKR theory) showed by a Griffith energy argument (assuming that contact over a circle of radius a introduces a surface energy −πa 2 ∆γ ) how the Hertz equations for the contact of elastic spheres are modifed by surface energy, and showed that the force needed to separate the spheres is equal to (3/2)πR∆γ, which is independent of the elastic modulus and so appears to be universally applicable and therefore to conflict with Bradley's answer.The discrepancy was explained by Tabor (1977), who identified a parameter µ ≡ R 1/3 ∆γ 2/3 /E * 2/3 ε governing the transition from the Bradley pull-off force 2πR∆γ to the JKR value (3/2)πR∆γ. Subsequently Muller et al. (1980) performed a complete numerical solution in terms of surface forces rather than surface energy, (combining the Lennard-Jones law of force between surfaces with the elastic equations for a halfspace), and confirmed that Tabor's parameter does indeed govern the transition.The numerical solution is repeated more accurately and in greater detail, confirming the results, but showing also that the load-approach curves become S-shaped for values of µ greater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of µ of 3 or more, but for low values of µ the simple Bradley equation better describes the behaviour under negative loads.

The Hertzian theory of elastic contact between spheres is extended by considering one of the spheres to be rough, so that contact occurs, as in practice, at a number of discrete microcontacts. It is found that the Hertzian results are valid at sufficiently high loads, but at lower loads the effective pressure distribution is much lower and extends much further than for smooth surfaces. The relevance to the physical-contact theory of friction and electric contact is considered.

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