Extended JKR theory on adhesive contact of a spherical tip onto a film on a substrate
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The conventional JKR theory was extended to the adhesive contact of a rigid sphere onto an elastic film perfectly bonded to a rigid substrate. An elasticity problem of axisymmetric indentation on an elastic film was revisited, in which the force–depth relations for both flat and spherical indentations were obtained in a simple form. With the obtained force–depth relations, the energy release rate at the debonding of a spherical tip from an elastic film was expressed in terms of pull-off force, elastic constants, and geometric parameters. The adhesion energy between a spherical tip and an elastic film can be measured as the critical energy release rate at the instability of debonding. This study suggests that when the critical radius of contact is larger than the thickness of an elastic film, the extended JKR theory should be used in place of the conventional JKR theory to correctly evaluate the adhesion energy between the spherical tip and the elastic film.
I. INTRODUCTION
The theory of adhesive contact between elastic bodies was first developed by Johnson, Kendall, and Robert,1 known as JKR theory, which is applicable to spherical indentation. Kendall2 meanwhile analyzed the work of adhesion of a flat punch to a plane surface. While these studies dealt with short-ranged surface interaction inside a contact area, Derjaguin, Muller, and Toporov3 used another approach known as DMT theory that considers long-ranged surface interaction only outside the contact area. Tabor4 clarified two extreme theories: the JKR theory is adequate for compliant materials with high surface energy and large radius, and the DMT theory is appropriate for stiff materials with low surface energy and small radius. Most polymers are in the JKR regime. Maugis5 applied the Dugdale-type cohesive-zone model to analyze the transition between the JKR and DMT regimes. Following these pioneering works, many studies have investigated the adhesion energy of contacting elastic bodies (refer to the book by Maugis6 and the review by Barthel7 for details). Adhesive contact between coated materials entails difficulties in obtaining the adhesion properties of the coating materials due to the effect of substrates. For example, if the JKR theory is used for spherical indentation onto an elastic film bonded to a substrate, only qualitative results are provided unless the film thickness is much larger than the contact radius of the spherical tip. Lebedev and Ufliand8 used the Henkel transform to solve a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2011.324 J. Mater. Res., Vol. 27, No. 1, Jan 14, 2012
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the problem of pressing a rigid stamp of a circular cross section onto an elastic layer supported (i.e., not bonded) on a rigid substrate without friction. Their solution technique was extended to the problem of flat indentation on an elastic film perfectly bonded to an elastic substrate.8–14 On the other hand, Mary et al.15 developed a semianalytical approach to adhesive contacts be
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