Extended JKR theory on adhesive contact of coated spheres

Thin films, coatings, and layered structures are ubiquitous in contemporary micro- and nanoelectronics, optoelectronics, and electromechanical systems, for which adhesion between heterogeneous materials is one of the important mechanical properties to be considered in fabrication and operation. In this study, the adhesive contact between elastic layered spheres, which includes a thin film on a flat substrate as a special case, is analyzed on the basis of the conventional Johnson–Kendall–Roberts (JKR) theory on adhesive contact between elastic spheres. Firstly, the force–depth relations of axisymmetric flat and spherical indentations on an elastic film perfectly bonded to an elastic half-space are obtained in compact forms, depending on two Dundurs parameters and the ratio of the contact radius to the thickness of the film. The solution of a spherical indentation on a layered half-space is then superposed, following a Hertz analysis, to obtain the adhesionless Hertzian contact between the elastic films coated on the elastic spheres. Similarly, the solution to the adhesive JKR contact state between elastic films coated on elastic spheres is also obtained by properly superposing the linear elastic solutions of the flat and spherical indentations on a layered half-space. The obtained Hertzian and adhesive JKR contact states should be regarded as approximate solutions, since the pressure distribution in the contact region does not exactly satisfy Newton’s third law. The JKR state equations among the normalized applied force, normalized penetration depth, and normalized contact radius are obtained in compact forms, in which non-dimensional multiplying factors depending on the geometric parameters and two sets of non-dimensional Dundurs parameters account for the effect of the elastic layers. Finally, the pull-off force at the moment of debonding of the two elastic films coated on the elastic spheres is calculated and compared with the conventional JKR result.