Adhesion of elastic spheres

Bradley (1932) showed that if two rigid spheres of radii R1 and R2 are placed in contact, they will adhere with a force 2πRΔγ, where R is the equivalent radius R1R2/(R1+R2) and A7 is the surface energy or 'work of adhesion' (equal to 71+72 712). 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)πRΔγ, 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, = R1/3Δ2/3/E* 2/3ε governing the transition from the Bradley pull-off force 2πRΔγ to the JKR value (3/2)π.RΔγ. 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 βgreater than one, leading to jumps into and out of contact. The JKR equations describe the behaviour well for values of /j, of 3 or more, but for low values of βthe simple Bradley equation better describes the behaviour under negative loads.