Rigorous Treatment of Contact Problems – Hertzian Contact

Abstract

In this chapter, a method is illustrated to find the exact solutions of contact problems in the framework of the “half-space approximation.” We examine, in detail, the classical contact problem of normal contact between a rigid sphere and an elastic half-space, which is often used to analyze more complex models.

As a preparatory step, we will summarize a few results of the theory of elasticity that have a direct application to contact mechanics. We consider the deformations in an elastic half-space, which are caused by a given stress acting upon its surface. The calculation of the deformation of an elastic body whose surface is being acted upon by a force (“direct problem of the theory of elasticity”) is much easier than the solution of contact problems, because in the latter, neither the stress distribution, nor the contact area are known to begin with. The classic solutions from Hertz (non-adhesive contact) and Johnson, Kendall, and Roberts (adhesive contact) use the known solutions for “direct problems” as building blocks to the construction of a solution for a contact problem.

An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-642-10803-7_21