This chapter concerns steady state problems in linear elasticity. This topic may appear to be the simplest in the whole of solid mechanics, but we will find that it offers many interesting mathematical challenges. Moreover, the material presented in this chapter will provide crucial underpinning to the more general theories of later chapters.
We will begin by listing some very simple explicit solutions which give valuable intuition concerning the role of the elastic moduli introduced in Chapter 1. Our first application of practical importance is elastic torsion, which concerns the twisting of an elastic bar. This leads to a class of exact solutions of the Navier equation in terms of solutions of Laplace's equation in two dimensions. However, as distinct from the use of Laplace's equation in, say, hydrodynamics or electromagnetism, the dependent variable is the displacement, which has a direct physical interpretation, rather than a potential, which does not. This means we have to be especially careful to ensure that the solution is single-valued in situations involving multiply-connected bars.
These remarks remain important when we move on to another class of two-dimensional problems called plane strain problems. These have even more general practical relevance but involve the biharmonic equation. This equation, which will be seen to be ubiquitous in linear elastostatics, poses significant extra difficulties as compared to Laplace's equation. In particular, we will find that it is much more difficult to construct explicit solutions using, for example, the method of separation of variables.