Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/837

813 ELASTICITY 813 substance is most strained at the ends of the smaller principal diameter, and least at the ends of the greater. In the equilateral triangular and square prisms there are longitudinal lines of maximum strain through the middles of the sides. In the oblong rectangular prism there are two lines of greater maximum strain through the middles of the broader pair of sides, and two lines of less maximum strain through the middles of the narrow sides. The strain is, as we may judge from the hydrokinetic analogy, ex cessively small, but not evanescent, in the projecting ribs of a prism of the figure shown in (2) of section 69. It is quite evanescent infinitely near the angle, in the triangular and rectangular prisms, and in each other case, us (5) of section 69, in which there is a finite angle, whether acute. or obtuse, projecting outwards. This reminds us of a general remark we have to make, although consideration of space may oblige us to leave it without formal proof. 72. Strain at Projecting Angles, evanescent ; at Re-entrant Angles, infinite; Liability to Cracks proceeding from Re entrant Angles, or any j)laces of too sharp concave curva ture. A solid of any elastic substance, isotropic or seolotropic, bounded by any surfaces presenting projecting edges or angles, or re-entrant angles or edges, however obtuse, cannot experience any finite stress or strain in the neighbourhood of a projecting angle (trihedral, polyhedral, or conical) ; in the neighbourhood of an edge, can only experience simple longitudinal stress parallel to the neigh bouring part of the edge ; and generally experiences infinite stress and strain in the neighbourhood of a re-entrant edge or angle ; when influenced by any distribution of force, exclusive of surface tractions infinitely near the angles or edges in question. An important application of the last part of this statement is the practical rule, well known in mechanics, that every re-entering edge or angle ought to be rounded, to prevent risk of rupture, in solid pieces designed to bear stress. An illustration of these principles is afforded by the concluding example of torsion in Thomson and Tait s section 707 ; in which we have the complete mathematical solution of the torsion problem for prisms of fan -shaped sections, such as the annexed forms (fig. 13). (i.) (3.) (4.) Fig. 13. The solution shows that when the solid is continuous from the circular cylindrical surface to its axis, as in (4), (5), (6), the strain is zero or infinite according as the angle between the bounding planes of the solid is less than or greater than two right angles as in cases (4) and (6) respectively. 73. Changes of Temperature produced by Compressions or Dilatations of a Fluid and Stresses of any kind in an Elastic Solid. From thermodynamic theory 1 it is concluded that cold is produced whenever a solid is strained by oppos ing, and heat when it is strained by yielding to, any elastic force of its own, the strength of which would diminish if the temperature were raised ; but that, on the contrary, heat is produced when a solid is strained against, and cold when it is strained by yielding to, any elastic force of its own, the strength of which would increase if the temperature were raised. When the strain is a condensa tion or dilatation, uniform in all directions, a fluid may be 1 W. Thomson on &quot; Thermo-elastic Properties of Matter,&quot; in Quarterly Journal of Mathematics, April 1855 (republishcd in Phil. Mag. 1377, second half year.) included in the statement. Hence the following propo sitions : (1.) A cubical compression of any elastic fluid or solid in an ordinary condition causes an evolution of heat; but, on the contrary, a cubical compression produces cold in any substance, solid or fluid, in such an abnormal state that it would contract if heated while kept under constant pressure. Water below its temperature (3 9 Cent.) of maximum density is a familiar instance. (See table of section 76.) (2.) If a wire already twisted be suddenly twisted further, always, however, within its limits of elasticity, cold will be produced ; and if it be allowed suddenly to untwist, heat will be evolved from itself (besides heat generated externally by any work allowed to be wasted, which it does in untwisting). It is assumed that the torsional rigidity of the wire is diminished by an elevation of temperature, as the writer of this article had found it to be for copper, iron, platinum, and other metals (compare section 78). (3.) A spiral spring suddenly drawn out will become lower in temperature, and will rise in temperature when suddenly allowed to draw in. [This result has been ex perimentally verified by Joule (&quot; Thermodynamic Proper ties of Solids,&quot; Trans. Roy. Soc., 1858) and the amount of the effect found to agree with that calculated, according to the preceding thermodynamic theory, from the amount of the weakening of the spring which he found by experiment.] (4.) A bar or rod or wire of any substance with or with out a weight hung on it, or experiencing any degree of end thrust, to begin with, becomes cooled if suddenly elongated by end pull or by diminution of end thrust, and warmed if suddenly shortened by end thrust or by diminu tion of end pull; except abnormal cases in which with constant end pull or end thrust elevation of temperature produces shortening; in every such case pull or diminished thrust produces elevation of temperature, thrust or di minished pull lowering of temperature. (5.) An india-rubber band suddenly drawn out within its limits of elasticity) becomes warmer; and when allowed to contract, it becomes colder. Any one may easily verify this curious property by placing an india-rubber band in slight contact with the edges of the lips, then suddenly extending it it becomes very perceptibly warmer : hold it for some time stretched nearly to breaking, and then suddenly allow it to shrink it becomes quite startingly colder, the cooling effect being sensible not merely to the lips but to the fingers holding the band. The first published statement of this curious observation is due to Gough (Memoirs of the Literary and Philosophical Society of Man chester, 2d series, vol. i. p. 288), quoted by Joule in his paper en &quot; Thermodynamic Properties of Solids &quot; (Transac tions of Royal Society, 1858). The thermodynamic con clusion from it is that an india-rubber band, stretched by a constant weight of sufficient amount hung on it, must, when heated, pull up the weight, and, when cooled, allow the weight to descend: this Gough, independently of thermo dynamic theory, had found to be actually the case. The experiment any one can make with the greatest ease by hanging a few pounds weight on a common india- rubber band, and taking a red-hot coal in a pair of tongs, or a red-hot poker, and moving it up and down close to the band. The way in which the weight rises when the red- hot body is near, and falls when it is removed, is quite startling. Joule experimented on the amount of shrinking per degree of elevation of temperature, with different weights hung on a band of vulcanized india-rubber, and found that they closely agreed with the amounts calculated by Thomson s theory from the heating effects of pull, and cooling effects of ceasing to pull, which he had observed in the same piece of india-rubber.