Elasticity is that property of matter which enables a body, whose form or bulk has been changed by force, to support without disintegration or further yielding the continued action of that force, and to recover its original form or bulk when left to itself. Any alteration in form or bulk of a body is called a strain, and the combination of forces producing the strain is called a stress. Now when an elastic body is strained, it becomes possessed of a certain power of doing work which it had not originally. As familiar examples of this we may mention the bent bow, the wound-up mainspring of a watch, and the compressed air of an air-gun. In these cases a certain amount of energy has to be spent to bring the substances into their strained condition. Then, on being allowed to recover themselves either suddenly or gradually by an appropriate withdrawal of the supporting stresses, these substances yield back, in the form of useful work, some of the energy originally expended. Again, in the phenomena of Sound (q.v.), we have constant illustrations of the elasticity of bodies. An elastic substance—a pianoforte string for example—is strained in a particular manner and then left to itself. It at once begins vibrating about its original position of equilibrium, the energy of strain being transformed into the energy of motion, and that again into the energy of strain, and so on until gradually it degenerates into heat under the influence of the viscosity of the air, the string itself, and whatever other elastic bodies may be in the vicinity.
Physically considered, sound is in fact a sub-section of elasticity. The power of conveying sound is possessed by all substances, whether solid or fluid; for such strains, then, all substances are truly elastic.
As regards fluids—i.e. gases and liquids, the only kind of elasticity that can exist is elasticity of bulk; for no portion of a fluid can sustain the action of a deforming force. Practically the only kind of stress we can apply to fluids so as to cause a change of volume is a pressure equal in all directions. It is found that at a given temperature, the volume of a given mass of fluid is determined by the pressure; so that, after being compressed by an increase of pressure, the fluid will recover its original volume when the pressure is brought back to its original value. Thus, all fluids are perfectly elastic. To discover how, under any circumstances, the pressure and volume are related to one another is clearly an elastic problem. Thus, we must regard Boyle's Law for gases, established by the Hon. R. Boyle (q.v.) for air in 1661, as the first experimental attempt to investigate the laws of elasticity (see GASES, and HYDRODYNAMICS). To the same class of problems belongs the determination of the compressibilities of liquids. Compression being defined as the diminution produced in unit volume of the substance by the application of a given extra pressure, compressibility is measured by the ratio of the compression to the related pressure, or, in more general language, by the ratio of the strain to the stress. Through considerable ranges of stress-values, this ratio is for most liquids constant. The reciprocal of the compressibility—i.e. the ratio of the appropriate stress and strain, is called the bulk-modulus of elasticity. It is the only modulus with which we have to do when the elasticity of fluids is being considered.
When we pass to the consideration of solids, we meet with another kind of elasticity, the elasticity of form. The resistance which a solid offers to a pure change of form, not involving a change of volume, is measured by its Rigidity (q.v.); and if for this change the body is truly elastic, the rigidity is measured by the ratio of the deforming stress to the resulting strain—i.e. it is the modulus of elasticity for this kind of stress and strain. A pure twist applied to a pillar, rod, or wire, is a strain which involves no change of volume. The resistance which such a pillar offers to torsion depends on its rigidity and form, but is independent of its compressibility. In most cases, however, when an elastic solid is strained by a particular kind of stress, the appropriate modulus of elasticity involves both the rigidity and the compressibility. Thus, when a wire is stretched by a longitudinal tension, it not only increases in length but also diminishes in section. The result is that any small cubical element becomes a brick-shaped portion somewhat greater in volume. It is usual to measure this kind of strain by the elongation simply, the contraction of the section being practically unimportant. The ratio of the tension to the elongation gives what is called Young's Modulus of Elasticity, a most important quantity in engineering. It is Young's modulus which also determines the resistance a beam or bar offers to bending. See STRENGTH OF MATERIALS.
The elasticity of solids is far from perfect—i.e. a very moderate stress well within the limits of rupture will produce a permanent set in the body, so that there will not be perfect recovery of form and dimensions when the stress is removed. It is difficult to fix accurately the so-called limits of perfect elasticity, since a small stress acting for a long time will produce a permanent set, which a much more powerful stress acting for a short time would not produce. So long, however, as the strains are small, experiment shows that the stresses are proportional to the corresponding strains. In other words, the modulus of elasticity is constant through considerable ranges of stress-values. This empirical law was first clearly enunciated by Hooke in 1678 in the words Ut Tensio sic Vis; and upon a generalised statement of Hooke's Law the whole modern theory of elasticity is based.
See Lord Kelvin's article 'Elasticity' in the Encyclopædia Britannica (separately published); Ibbetson's Mathematical Theory of Elasticity (1887); Love's Mathematical Theory of Elasticity (2 vols. 1890-93); and, for a valuable compendium and criticism of the labours of other elasticists, the History of the Elasticity and Strength of Materials (Cambridge University Press, 2 vols. 1886-94), by Todhunter and Pearson; also the articles in this work on STRAIN, STRESS, STRENGTH OF MATERIALS, TORSION.