The JKR formalism in applications to problems of adhesive contact

The JKR (Johnson-Kendall-Roberts) problem of adhesive contact between elastic spheres is an example of a mathematically beautiful theory that has many practical applications. However, it would be erroneous to reduce the JKR theory to just the problem of adhesive contact between spheres. Indeed, simultaneously with the presentation of the JKR theory, Kendall (J Phys D Appl Phys 4:1186–1195, 1971) applied the equilibrium theory of adhesion to bodies of other geometries and coatings. It gives a review of applications of the JKR formalism to axisymmetric indenters of various shapes, various elastic materials, different conditions of contact, and elastic structures. These structures include thin and thick elastic layers and atomically thin stretched membranes. The JKR formalism means that an adhesive contact problem may be solved by combining two ideas: (1) the Derjaguin balance energy approach (Derjaguin, Kolloid Z 69:155–164, 1934) and (2) superposition of solutions to two non-adhesive contact problems (the Hertz-type and the Boussinesq-type problems). The JKR formalism may be used if the distance between the free surface of the material and the indenter surface increases rapidly at the periphery of the contact region and the solutions of two contact problems having the same contact area may be superimposed on each other. It is shown that the JKR formalism may be reinforced if one employs the properties of slopes of the force-displacement diagrams of non-adhesive indentation. For the first time, such reinforcements were demonstrated explicitly by the author (Borodich, Adv Appl Mech 47:225–366, 2014). It is argued that the JKR formalism may be applied to an enormous number of adhesive contact problems for various elastic structures