Introduction

This chapter concerns simple unsteady problems in linear elasticity. As noted in Section 1.10, the unsteady Navier equation (1.7.8) bears some similarity to the familiar scalar wave equation governing small transverse displacements of an elastic string or membrane. We therefore start by reviewing the main properties of this equation and some useful solution techniques. This allows us to introduce, in a simple context, important concepts such as normal modes, plane waves and characteristics that underpin most problems in linear elastodynamics.

In contrast with the classical scalar wave equation, the Navier equation is a vector wave equation, and this introduces many interesting new properties. The first that we will encounter is that the Navier equation in an infinite medium supports two distinct kinds of plane waves which propagate at two different speeds. These are known as P-waves and S-waves, and correspond to compressional and shearing oscillations of the medium respectively. Considered individually, both P- and S-waves behave very much like waves as modelled by the scalar wave equation (1.10.9). In practice, though, they very rarely exist in isolation since any boundaries present inevitably convert P-waves into S-waves and vice versa. We will illustrate this phenomenon of mode conversion in Section 3.2.5 by considering the reflection of waves at a plane boundary.

In two-dimensional and axisymmetric problems, we found in Chapter 2 that the steady Navier equation may be transformed into a single biharmonic equation by introducing a suitable stress function. In Sections 3.3 and 3.4 we will find that the same approach often works even for unsteady problems, and pays dividends when we come to analyse normal modes in cylinders and spheres.