Royal Society Publishing

Adhesion of elastic spheres

J. A. Greenwood


Bradley (1932) showed that if two rigid spheres of radii R1 and R2 are placed in contact, they will adhere with a force 2πΔRγ, where R is the equivalent radius R1R1/(R1+R2) and Δγ is the surface energy or ‘work of adhesion’ (equal to γ1+γ2-γ12). 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 -πa2Δγ) 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)πΔRγ, 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 3Δγ2/3/E*2/3\ε governing the transition from the Bradley pull-off force 2πRΔ|γ to the JKR value (3/2)πRΔγ. 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 half-space), 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.