An upper bound to multiscale roughness-induced adhesion enhancement

Recently Guduru and coworkers have demonstrated with neat theory and experiments that both increase of strength and of toughness are possible in the contact of a rigid sphere with concentric single scale of waviness, against a very soft material. The present note tries to answer the question of a multiscale enhancement of adhesion, considering a Weierstrass series to represent the multiscale roughness, and analytical results only are used. It is concluded that the enhancement is bounded for low fractal dimensions but it can happen, and possibly to very high values, whereas it is even unbounded for high fractal dimensions, but it is also much less likely to occur, because of separated contacts.