Rigorous Treatment of Contact Problems – Adhesive Contact

Abstract

The problem of normal contact (without adhesion) between elastic bodies with slightly curved surfaces was solved in 1882 by Hertz. Bradley presented the solution 50 years later for adhesive normal contact between a rigid sphere and a rigid plane. The resulting adhesive force was found to be F A = 4π γ R, where γ is the surface energy. The solution for the adhesive contact between elastic bodies was presented in 1971 by Johnson, Kendall, and Roberts (JKR-Theory). They obtained F A = 3π γ R for the adhesive force. Derjaguin, Muller, and Toporov published an alternative adhesive theory in 1975, which is known as the DMT-Theory. After an intense discussion in 1976 Tabor came to the realization that the JKR-Theory and the DMT-Theory are both correct and are special cases of the general problem. For absolutely rigid bodies the theory from Bradley is valid; for small, rigid spheres DMT-Theory is valid; and for large, flexible spheres JKR-Theory. The difference between all of these cases, however, is very minor and the JKR-Theory describes adhesion relatively well, even in the scope of the DMT-Theory. Perhaps this is why JKR-Theory is used so prevalently to describe adhesive contacts. For this reason, we will also limit ourselves in this chapter to the presentation of the theory from Johnson, Kendall, and Roberts.