Chapter 2
Contact of rough surface: a literature survey
2.1. Basics of Contact Mechanics
2.1.1 Introduction
Contact phenomena are abundant in everyday life and play a very important role in engineering structures and systems. They include friction, wear, adhesion and
lubrication, among other things; are inherently complex and time dependent; take place on the outer surfaces of parts and components, and involve thermal, physical and
chemical processes. Contact Mechanics is the study of relative motion, interactive forces and tribological behavior of two rigid or deformable solid bodies which touch or
rub on each other over parts of their boundaries during lapses of time. However, the contact between deformable bodies is very complicated and it is not yet well
understood. The contact theory was originally developed by Hertz [1] and it remains the
foundation for most contact problems encountered in engineering. It applies to normal contact between two elastic solids that are smooth and can be described locally with
orthogonal radii of curvature such as a toroid. Furthermore, the size of the actual contact area must be small compared to the dimensions of each body and to the radii of
curvature non-conforming contact. Hertz made the assumption based on observations that the contact area is elliptical in shape for such three dimensional bodies. The
equations simplify when the contact area is circular such as with spheres in contact. At extremely elliptical contact, the contact area is assumed to have constant width over the
length of contact such as between parallel cylinders. The Hertz theory is restricted to frictionless surfaces and perfectly elastic solids. It was not until nearly one hundred
years later that Johnson, Kendall, and Roberts [2] found a similar solution for the case of adhesive contact. This theory was rejected by Boris Derjaguin and co-workers [3]
who proposed a different theory of adhesion in the 1970s. The Derjaguin model came to be known as the DMT after Derjaguin, Muller and Toporov model, and the Johnson et
al. model came to be known as the JKR after Johnson, Kendall and Roberts model for
adhesive elastic contact. This rejection proved to be instrumental in the development of the Tabor [4] and later Maugis [5] parameters that quantify which contact model of the
JKR and DMT models represent adhesive contact better for specific materials. Further advancement in the field of contact mechanics in the mid-twentieth
century may be attributed to names such as Bowden and Tabor [58]. They were the first to emphasize the importance of surface roughness for bodies in contact. Through
investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area. Such understanding also drastically
changed the direction of undertakings in tribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.
The contributions of Archard [6] must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces,
the contact area is approximately proportional to the normal force. Further important insights along these lines were provided by Greenwood and Williamson [7], Bush
[8], and Persson [9]. The main findings of these works were that the true contact surface in rough materials is generally proportional to the normal force, while the parameters of
individual micro-contacts i.e. pressure, size of the micro-contact are only weakly dependent upon the load.
2.1.2 Elastic Contact
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