Introduction

So far in this book we have been considering linear elasticity only for very simple geometries such as cylinders, spheres and half-spaces. In this chapter, we will consider more general solids under the restriction that they are thin and the equations of elasticity can consequently be simplified. A familiar example that we have already encountered is the wave equation governing the transverse displacements of a thin elastic string, and we will revisit this model below in Section 4.3.

A string is characterised by its inability to withstand any appreciable shear stress, so its only internal force is a tension acting in the tangential direction. Similarly, a membrane is a thin, nearly two-dimensional structure, such as the skin of a drum, which supports only in-plane tensions. A bar, on the other hand, is a nearly one-dimensional solid that can be subject to either tension or compression. However, many thin elastic bodies also have an appreciable bending stiffness and therefore admit internal shear stress as well as tension. A familiar example is a flexible ruler, which clearly resists bending while deforming transversely in two dimensions, and is known as a beam. A thin, nearly one-dimensional object which can bend in both transverse directions, such as a curtain rod or a strand of hair, will be referred to as a rod. On the other hand, a nearly planar elastic structure with significant bending stiffness, for example a pane of glass or a stiff piece of paper, is called a plate. Finally, a shell is a thin, nearly two-dimensional elastic body which is not initially planar, for example a ping-pong ball or the curved panel of a car.