On the dynamic JKR adhesion problem

Shui et al. (2020) have recently shown that applying high-frequency vibrations, we can increase the mean adhesion between viscoelastic solids. This is due to the fact that oscillating contact area leads to an effect of increased apparent surface energy during the retraction phase which can be described by the well known empirical Gent and Schultz law (GS). However, Shui et al solution surprisingly appears not to depend on GS constants, which would imply perhaps no amplification. Yi et al. (2024) have made similar experiments, and proposed a simpler fitting model, which seems to work however with widely different GS constant when changing the sphere radius. Here, we solve the JKR dynamic adhesion problem for a sphere oscillating on a substrate by imposing an harmonic oscillation of the contact area, which permits to obtain a very simple solution by simply averaging the resulting cycle of indentation. We find that the solution is close to a JKR form for the mean indentation vs mean force, which we find in a simple approximation. Although there is saturation in the amplification when the contact radius shrinks to zero and the problem becomes that of impacts at large amplitudes of vibrations, experiments show that other saturations occurs first, presently unclear. We discuss also the influence of resonances. We find reasonable agreement with experiments conducted on PDMS.