Chapter XIII - ELASTICITY

Summary

13·1. So far we have been dealing largely with such mathematical fictions as rigid bodies, weightless rods or inextensible strings, but in the present chapter we propose to bring our investigations into closer touch with reality by showing how to make allowance for the fact that bodies are not rigid but undergo small changes in form when subject to the action of force.

We shall confine our considerations to a few simple cases of isotropic bodies. An isotropic body is such that if a sphere is cut out of the body anywhere it possesses no directional properties of any kind, as distinct from a crystalline body or a body of fibrous structure.

The simplest type of deformation or strain that a body can undergo is a uniform extension, in which all elements of length PQ in a certain direction are altered to elements of length P′Q′ such that the increment in length P′Q′ – P′Q′ is a certain fraction ∈ of the original length, i.e. ∈=(P′Q′ – P′Q′)|PQ. This fraction e is then called the extension.

Regarding a contraction as a negative extension it is clearly possible for a body to be extended in more directions than one. It is obvious, for example, that if a bar is extended longitudinally it will in general contract laterally. We do not propose, however, to analyse the different kinds of strain that are possible but only to deal with some simple cases.