22. The longitudinal extension of the bar under tension is not the only strain in the bar. It is accompanied by a lateral contraction by which all the transverse filaments of the bar are shortened. The amount of this contraction is [sigma]T/E, where [sigma] is a certain number called _Poisson's ratio_, because its importance was at first noted by S.D. Poisson in 1828. Poisson arrived at the existence of this contraction, and the corresponding number [sigma], from theoretical considerations, and his theory led him to assign to [sigma] the value ¼. Many experiments have been made with the view of determining [sigma], with the result that it has been found to be different for different materials, although for very many it does not differ much from ¼. For steel the best value (Amagat's) is 0.268. Poisson's theory admits of being modified so as to agree with the results of experiment.
23. The behaviour of an elastic solid body, strained within the limits of its elasticity, is entirely determined by the constants E and [sigma] if the body is _isotropic_, that is to say, if it has the same quality in all directions around any point. Nevertheless it is convenient to introduce other constants which are related to the action of particular sorts of forces. The most important of these are the "modulus of compression" (or "bulk modulus") and the "rigidity" (or "modulus of shear"). To define the _modulus of compression_, we suppose that a solid body of any form is subjected to uniform hydrostatic pressure of amount p. The state of stress within it will be one of uniform pressure, the same at all points, and the same in all directions round any point. There will be compression, the same at all points, and proportional to the pressure; and the amount of the compression can be expressed as p/k. The quantity k is the modulus of compression. In this case the linear contraction in any direction is p/3k; but in general the linear extension (or contraction) is not one-third of the cubical dilatation (or compression).
24. To define the _rigidity_, we suppose that a solid body is subjected to forces in such a way that there is shearing stress within it. For example, a cubical block may be subjected to opposing tractions on opposite faces acting in directions which are parallel to an edge of the cube and to both the faces. Let S be the amount of the traction, and let it be uniformly distributed over the faces. As we have seen (§ 7), equal tractions must act upon two other faces in suitable directions in order to maintain equilibrium (see fig. 2 of § 7). The two directions involved may be chosen as axes of x, y as in that figure. Then the state of stress will be one in which the stress-component denoted by X_y is equal to S, and the remaining stress-components vanish; and the strain produced in the body is shearing strain of the type denoted by e _(xy). The amount of the shearing strain is S/µ, and the quantity µ is the "rigidity."