Adhesion of elastic spheres

Abstract
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.

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