Adhesion of Spheres JKR Theory

The Hertz theory allows to calculate the contact shape and forces between spheres under the influence of an external force. It does not include any surface force and therefore does not lead to an expression for the adhesion force. When moved apart, the bodies separate at the point where 6 = 0 and o = 0 without any adhesion force. A first model to include adhesive forces based on the Hertz theory was introduced by Derjaguin in 1934 [84]. He assumed that the contact shape is that given by the Hertz theory and that the total energy of the system is the elastic energy as given by the Hertz model minus the energy due to the formation of the contact area na. In Derjaguin s model, the force to achieve a certain indentation 6 is reduced by 

An extension of the Hertz theory taking adhesive interactions and their influence on the contact shape into account was introduced in 1971 by Johnson, Kendall and Roberts [852] and it has become well known as the JKR theory. Their basic assumption was to take into account adhesive interaction only within the contact zone and neglect any interactions outside the contact zone. 

When loading the contact up to the force F in the absence of adhesion (iva = 0), we follow the Hertzian P(8) curve from 0 to A. At A, we have the contact radius a and the indentation 8i = a /R and the stored elastic energy is (Eq. (8.47)) 

The inner circle of the contact area will be under compressive stress, whereas outer annular zone will be under tensile stress. The radius of the inner compressed zone 

The tensile stresses go to infinity at the rim of the contact, but the total stress integrated over the whole contact area remains finite. Outside the contact area, the vertical stress is zero since the JKR model assumes that no surface forces act outside the contact area. The infinite stresses at the edge of the contact are physically not possible. Obviously, the description of that region down to molecular scales by a continuum model as the JKR theory carmot be realistic. As soon as one assumes realistic interaction potentials between the molecules at the rim of the contact, these singularities will disappear. However, such detailed models will usually be too complicated to allow their routine use in contact mechanics. 

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