In this chapter we discuss applications of the perturbation approach to stability analysis of elastic bodies subjected to large deformations. Various ideas commonly used in the perturbation approach are explained by using simple examples. Two types of bifurcations are distinguished: bifurcations at a non-zero critical mode number and bifurcations at a zero critical mode number. For each type we first explain with the aid of a model problem how stability analysis can be carried out and then explain how the analysis could be extended to problems in Finite Elasticity. Although the present analysis focuses on the perturbation approach, the dynamical systems approach is also discussed briefly and references are made to the literature where more details can be found. In the final section, we carry out a detailed analysis for the necking instability of an incompressible elastic plate under stretching.

Introduction

This chapter is concerned with nonlinear stability analysis of elastic bodies subjected to large elastic deformations. A typical problem we have in mind is the stability of a cylindrical rubber tube that is compressed either by an external pressure or by forces at the two flat ends. In general terms, we consider an elastic body which has an undeformed configuration Br in a three-dimensional Euclidean point space. This elastic body is then subjected to some external forces. It is now customary to refer to such an elastic body as pre-stressed in the Finite Elasticity literature (the pre-stress considered in the present context is not therefore that induced in a manufacturing process).