ELASTICITY. Every solid, no matter how rigid we may think it to be, will have its dimensions changed upon the application of force. If the resulting distortion does not ex ceed a certain amount it will disappear when the force is removed. Bodies which recover from the distorting effect of force and resume their original configurations are said to be elastic. The relation between force and the deformation it produces is studied under the title elasticity; the harmful effects of distor tion and the proportioning of materials safely to resist given forces belong to the strength of materials (q.v.). The behavior of gases comes under thermodynamics, of liquids under hydro dynamics. In the mechanics of rigid bodies forces are represented by arrows placed at points; actually, they are distributed over sur faces. We are concerned here with the area of action as well as with the magnitude of the force, and shall therefore use the word stress to denote force per unit area. Stress, con trary to this usage, is generally regarded as synonymous with force, and what we here call stress is commonly called °intensity of stress' or "unit stress.' Stress as defined in this arti cle is not force; its dimensions are or and the unit is the pound or ton per square inch and the dyne and kilogram per square centimeter. A force oblique to a sur face can be resolved into normal and tangential components. The resulting normal stress is ac companied by change of length. The length per unit length is called linear strain ; there are likewise areal and volumetric strains. Strain is a pure number without dimensions. Tangential stress or shear stress produces an gular distortion due to the sliding of one layer of material with respect to the adjacent layers. For instance, if the two covers of a book are shifted parallel to each other there will he relative sliding of the leaves. Any straight line drawn on the top or bottom end of the book will change its inclination; the change of a right angle is called shear strain. It is the prov ince of the theory of elasticity to investigate mathematically the consequences which result from an experimentally found relation between stress and strain. The first experiments — on the rupture of beams—were made by Galileo, e Dimostrazioni matematiche) (1638). His results were of no value because he supposed the fibres of a beam to be inexten sible, yet his work was the impulse to subse quent inquiries. It was not until 1678 that any relation between stress and strain was pub lished. In that year Hooke in his (De potentia restitutiva,) announced the law known by his name in the form of an anagram ceiiinosssttuu containing the letters of Ut tensio sic via, i.e., the force varies directly as the extension. He
claimed to have discovered it in 1660. Until the end of the 18th century only special problems on beams, columns, and plates were attacked; this period was almost barren of experimental work. The foundation of the mathematical theory was laid by Navier, (Metnoire sur les lois . . . des corps solides elastiques,) Mintoires de tinstitut, Vol. VII, which was read to the Academie des Sciences in 1821. Progress was rapid after this in the hands of such masters as Cauchy, Clapyron, Green, Lame and Poisson, and culminated in the life long labors of Barre de Saint-Venant (1797 1886). For the detailed history of the subject through the time of Saint-Venant consult Tod hunter and Pearson, of the Theory of Elasticity and of the Strength of Materials) (Cambridge 1886) ,• subsequent investigations are noted in the introduction to Love, Mathe matical Theory of Elasticity) (2d ed., Cam bridge 1906).
Strain. Before taking up the relations be tween stress and strain, we shall studithe small displacements suffered by •an infinitesimal ele ment dx dy ds within a medium in any state of stress. shows the projections of two concurrent edges on the xy plane before and after displacement; the yz and sx diagrams are omitted for brevity. Let (x, y, s), the cor ner nearest the origin, be displaced u, v, w, where u, v, w are small compared with x, y, z. Then, to terms of the first order, the ends of dx and dy receive the axial displacements shown. For if a variable increases infinites imally, the function will increase differen tially; thus if the left end of dx, distant x from the origin, moves u parallel to x, a point infinitesimally further from the origin will move ,ve infinitesimally more, ie., the ax derivative being partial to indicate that the crement was due only to a change of x. The x-projection of the elongated length of dx is dx + dx, which, since x is infinitesimal, is ax itself the new length of dx. Hence the stretch au of dx is dx and if the linear strain at x, y, oe is denoted by ens au Oo Ow axe — ems= — ax ay as (1)the other two components being derived by cyclic permutations of the letters. By defi nition the shear strains are the decrements of the right angles formed by the concurrent edges at (x, y, z). If they are denoted by eyx, ear in the co-ordinate planes it is evi dent from Fig. 1 that = A- au. Since is infinitesimal by hypothesis, it equals its tangent — as Bfewise ati exiav au ax aw av as au aw (2) as as' The six quantities exy are the components or constituents of strain at the point (x, y, z). The shear strain suffered by an element rotates it as a whole, the amount being measured by the rotation of its diagonal.