Description

The elastic-plastic contact of spheres under combined normal and tangential loading is a fundamental problem in contact mechanics. It is applicable, for example, in particle handling @1#, or friction between contacting rough surfaces @2#. It can be found in a variety of technologies, both new and traditional as described by Tichy and Meyer @3#. Indeed, the number of works on spherical contact under steady state sliding ~see reviews by Bhushan @4# and by Adams and Nosonovsky @5#! and under static or rolling conditions ~see review by Johnson @6#! that were published so far is impressive. Issues such as normal frictional loading, tangential loading and sliding inception of contacting elastic bodies are well described @7#. However, in spite of their elastic-plastic nature, the mathematical complexity involved with these frictional contact problems @8# restricted their solutions to mostly linear elasticity. Combined loading begins with a normal component that produces the contact area, which can then support an added tangential component and, hence, should be treated as a frictional contact problem.

Frictional normal loading of elastic spheres differs from the classical frictionless contact problem of the Hertz theory ~e.g., @9#! and was treated by several researchers, @10–13#. However, when the contacting sphere materials have the same elastic constants ~or are incompressible! the mutual contact pressure produces identical tangential displacements that eliminate any tendency of interfacial slip. Hence, no shear stress is developed in the interface and the Hertz theory is still valid. Johnson @6# concluded that even with dissimilar realistic materials the local shear traction is an order of magnitude smaller than the corresponding local contact pressure and, hence, may be neglected ~except for brittle materials!. Historically, the treatment of combined loading of spherical contact started with the pioneering works by Mindlin @14# and Mindlin and Deresiewicz @15# ~in fact an earlier paper from 1938 in Italian is due to Cattaneo, see @7#!. They showed that the contact of two spheres under combined loading consists of a central stick region surrounded by a slip zone. As the tangential force increases, the size of the stick region gradually vanishes until full sliding begins when the Coulomb friction law is satisfied. The normal loading was assumed to be frictionless and the resulting contact area and contact pressure distribution remained these of the Hertz theory, even when the tangential force was added.