Simple models of viscoelastic fibrillar adhesion to a rigid sphere

The SPG (Schargott–Popov–Gorb) model of independent elastic fibrils in adhesive contact with a rigid sphere is a simple model of bioinspired adhesives, which with remarkable simplicity explains the preload dependence of pull-off and saturation when the compressive force is sufficiently high. However, typical materials of fibrils are viscoelastic, so we extend the model to consider viscoelastic behavior using a standard material model for each fibril. We use for the condition of nucleation of fibril detachment either the classical Schapery model or a more recent proposal by Shrimali and Lopez-Pamies. In either case, starting from a relaxed compressive contact, we find that the pull-off force also strongly depends on preload and is amplified by faster unloading rates, very differently for the two theories. A propagation theory follows for the Schapery model which seems to link the effective work of adhesion on contact retraction speed where the characteristic velocity has however nothing to do with the classical Schapery propagation theory, whereas the Shrimali and Lopez-Pamies model leads to detachment without a real prior phase of propagation. Some results are compared with those in the literature.