A Maugis-Dugdale cohesive solution for adhesion of a surface with a dimple

We study the adhesion of a surface with a 'dimple' which shows a mechanism for a bi-stable adhesive system in surfaces with spaced patterns of depressions, leading to adhesion enhancement, high dissipation and hysteresis. Recent studies were limited mainly to the very short range of adhesion (the so-called JKR regime), while we generalize the study to a Maugis cohesive model. A 'generalized Tabor parameter', given by the ratio of theoretical strength to elastic modulus, multiplied by the ratio of dimple width to depth has been found. It is shown that bistability disappears for generalized Tabor parameter less than about 2. Introduction of the theoretical strength is needed to have significant results when the system has gone in full contact, unless one postulates alternative limits to full contact, such as air entrapment, contaminants or fine scale roughness. Simple equations are obtained for the pull-off and for the full contact pressure in the entire set of the two governing dimensionless parameters. A qualitative comparison with results of recent experiments with nanopatterned bioinspired dry adhesives is attempted in light of the present model.