Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/843

819 E L A S T I C I T Y 819 1350 turns of permanent twist, when the diminution had amounted to 14 per cent, of the primitive value, 36 turns more broke the wire before another determination of torsionul rigidity had been made. The steel pianoforte wire also showed a diminution of tursional rigidity with permanent twist, and (as did the copper wire) showed first a diminution and then a slight augmentation. The amount of the diminution in the steel wire was enormously greater than the surprisingly great amount which had been discovered in the copper wire, and the ultimate augmentation was considerably greater in the steel than what it had been in the copper before rupture. Thus after 473 turns of permanent twist the torsional modulus had diminished from 751 million grammes per square centimetre to 414 ! 95 more turns of permanent twist augmented the rigidity from 414 to 430, and when farther twisted the wire broke before another observation had been made. The vibrator used in these experiments was a cylinder of lead weighing 56 Ib, which was kept hanging on the wire while it was being twisted, and in fact during the whole of about 100 hours from the beginning of the experiment till the wire broke, except on two occasions for a few minutes, w T hile the top fastening which had given way was being resoldered. The period of vibration was augmented from 39 375 seconds to 51 9 seconds by the twist. The wire took the twist very irregularly, some parts not beginning to show much signs of permanent twist till near the end of the experiment In two specimens of copper wire of the same length and gauge as those described above, the Young s modulus was found to be increased 10 per cent, by 100 turns of permanent twist. Five metres of the steel pianoforte wire, bearing a weight of 39 Ib, was in one of Mr M Farlane s experiments twisted 120 turns, and then allowed to untwist, and 38^ turns came out, leaving the wire in equilibrium with 81 f turns of permanent twist. Its Young s modulus was then found not to differ as much as ^ per cent, from the value it had before the wire was twisted. MATHEMATICAL THEOEY OF ELASTICITY. 1 PART I. ON STRESSES AND STRAINS. 2 CHAPTER I. Initial Definitions and Explanations. Def. A stress is an equilibrating application of force to a body. Cor. The stress on any part of a body in equilibrium will thus signify the force which, it experiences from the matter touching that part all round, whether entirely homogeneous with itself, or only so across a portion of its bounding surface. Dcf. A strain is any definite alteration of form or dimensions experienced by a solid. Examples. Equal and opposite forces acting at the two ends of a wire or rod of any substance constitute a stress upon it. A body pressed equally all round for instance, any mass touched by air on all sides experiences a stress. A stone in a building experiences stress if it is pressed upon by other stones, or by any parts of the structure, in contact with it. Any part of a continuous solid mass, simply resting on a fixed base, experiences stress from the surrounding parts in consequence of their weight. The different parts of a ship in a heavy sea experience stresses from which they are exempt when the water is smooth. If a rod of any substance become either longer or shorter, it is said to experi ence a strain, if a body be uniformly condensed in all directions it experiences a strain. If a stone, a beam, or a mass of metal in a building, or in a piece of framework, becomes condensed or dilated in any direction, or bent, or twisted, or distorted in any way, it is said to experience a strain, to become strained or often in common language, simply &quot; to strain.&quot; A ship is said to &quot; strain &quot; if in launching, or when working in a heavy sea, the different parts of it experience relative motions. CHAPTER II. Homogeneous Stresses and Homogeneous Strains. Dcf. A stress is said to be homogeneous throughout a body when equal and similar portions of the body, with corresponding lines theory of Waves in an reolotropic or isotropic elastic solid, is new. 2 These terms were first definitively introduced into the Theory of Elasticity by Rankine, and have been found very valuable in writing on the&quot; subject. It wiil parallel, experience equal and parallel twssures or tensions on cor responding elements of their surfaces. Cor. When a body is subjected to any homogeneous stress, the mutual tension or pressure between the parts of it on two sides of any plane amounts to the same per unit of surface as that between the parts on the two sides of any parallel plane ; and the former tension or pressure is parallel to the latter. A strain is said to be homogeneous throughout a body, or the body is said to be homogeneously strained, when equal and similar portions, with corresponding lines parallel, experience equal and similar alterations of dimensions Cor. All the particles of the body in parallel pianes remain in par allel planes, when the body is homogeneously strained in any way. Examples. A long uniform rod, if pulled out, or a pillar loaded with a weight, will experience a uniform strain, except near its ends. There will be a sensible heterogeneousness of the strain, because of the end attachments, or other cir cumstances preventing the ends from expanding laterally to the same extent as the middle does. A piece of cloth held in a plane, and distorted so that a warp and woof, instead of being perpendicular to one another, become two sets of parallels cutting one another obliquely, experiences a homogeneous strain. The strain is heteroge neous as to intensity, from the axis to the surface of a cylindrical wire under torsion, mid heterogeneous as to direction in different positions in a circle round the axis. CHAPTER III. On the Distribution oj Force in a Stress. Theorem. In every homogeneous stress there is a system of three rectangular planes, each of which is perpendicular to the direction of the mutual force between the parts of the body on its two fides. For let P(X), P(Y), P(Z) denote the components, parallel to X, Y, Z, any three rect angular lines of reference, of the force experienced per unit of surface at any por tion of the solid bounded by a plane parallel to (Y, Z) ; Q(X), Q(Y), Q(Z), the corre sponding components of the force experienced by any surface of the solid parallel to(Z, X);and K(X), K(Y), K(Z), those of the force at a surface parallel to (X, Y). Now, by considering the equilibrium of a cube ot the solid with faces parallel to the planes of reference (fig. 15), we see that the couple of forces Q(Z) on its two faces perpendicular to Y is balanced by the couple of forces K(Y) on the faces perpendicular to Z. Hence we must have R(Y) Q(Z) =R(Y). R(X) = F(Z) Similarly it is seen that and P(Y)=rQ(X). For the sake of brevity, these pairs of equal quantities (being tan gential forces respectively perpendicular to X, Y, Z) may bo denoted by T(X), T(Y), T(Z). Consider a tetrahedral portion of the body (surrounded it may be with continuous solid) contained within three planes A, B, C, through a point parallel to the planes of the pairs of lines of reference, and a third plane K cutting these at angles a, /3, y respectively ; so that as regards the areas of the different sides we shall have A = K cos a, B = K cos /?, C = K cos y. The forces actually experienced by the sides A, B, C have nothing to balance them except the force actually experienced by K. Hence those three forces must have a single resTiltant, and the force on K must be equal and opposite to it. If, therefore, the force on K per unit of surface be denoted by F. and its direction cosines bv I, m, n, we have F.K.Z F.K. m F.K.n =T(Y)A + T(X)B + R(Z)C; and, by the relations between the cases stated above, we deduce Fi =P(X) cosa + T(Z) cos/3 + T(T) cosy Fm = T(Z) cosa + Q(Y) cos/3 + T(X) cos y F/t =T(Y) cos a + T(X) cos/3 + R(Z) cos y Hence the problem of finding (a, 0, y), so that the force F (I, m, n) may be perpendicular to it, will be solved by substituting cos o, cos f), cos y for I, m, n in these equations. By the elimination of cos o, cos 0, cos y from the three equations thus obtained, we have the well-known cubic determinants equation, of which the roots, necessarily real, lead, when no two of them are equal, to one and only one system of three rectangular axes having the stated property.