Maugls-Tabor parameter dependence of pull-off in viscoelastic fine Hertzian contacts

We derive an extension of the Maugis-Dugdale-Johnson-Greenwood model for 2D adhesive Hertzian contact to viscoelastic materials. This results in extremely simple approximate results for the maximum amplification of the pull-off due to viscoelastic effects, for arbitrary form of the viscoelastic linear properties. In particular, we assume that the initial loading state is fully relaxed, and unloading occurs at very large pulling speeds. We show that the maximum amplification (of the order of (E-I/E-R)(2/3), the ratio between instantaneous and relaxed modulus to the 2/3 power) is only reached for large and soft cylinders, namely, large enough (relaxed) Tabor-Maugis parameters lambda of the order of the order of E-I/E-R, and therefore typically much larger than the Tabor-Maugis parameters to have a "short-range" JKR adhesion regime in quasi-static conditions, which is lambda similar to 1. The results agree well with recent numerical ones by Muser-Persson, but there is an shift factor in the Tabor-Maugis parameter which requires further study and may depend on initial contact area.