814 ELASTICITY 74. The thermodynamic theory gives one formula 1 by which the change of temperature in every such case may be calculated when the other physical properties are known : fi- tep - where 6 denotes the elevation of temperature produced by the sudden application of a stress p ; t, the temperature of the substance on the absolute thermodynamic scale, 2 the change of temperature being supposed to be but a very small fraction of t ; e, the geometrical effect (expansion or other strain) produced by an elevation of temperature of one degree when the body is kept under constant stress ; K, the specific heat of the substance per unit mass under constant stress ; p, the density; and J, Joule s equivalent (taken as 42400 centimetres). In using the formula for a fluid, p must be normal pressure equal in all directions, or normal pressure on a set of parallel planes, or tangential traction on one or other of the two sets of mutually perpendicular parallel planes which (section 43) experience tangential traction when the body is subjected to a simple distorting stress ; or, quite generally, p may be the proper numerical reckoning (Mathematical Theory, chap, x.) of any stress, simple or compound. When p is pressure uniform in all directions, e must be expansion of bulk, whether the body expands equally in all directions or not. When p is pressure per pendicular to a set of parallel planes, e must be expansion in the direction opposed to this pressure, irrespectively of any change of shape not altering the distance between the two planes of the solid perpendicular to the direction of p. When p is a simple tangential stress, reckoned as in section 43, e must be the change, reckoned in fraction of the radian, of the angle, infinitely nearly a right angle, between the two sets of parallel planes in either of which there is the tangential traction denoted by p. In each of these cases p is reckoned simply in units of force per unit of area. Quite generally p may be any stress, simple or compound, and e must be the component (Math. Th., chaps, viii. and ix.) relatively to the type of p, of the strain produced by an elevation of temperature of one degree when the body is kept under constant stress. The constant stress for which K and e are reckoned ought to be the mean of the stresses which the body experiences with and without p. Mathe matically speaking, p is to be infinitesimal, but practically it may be of any magnitude moderate enough not to give any sensible difference in the value of either K or e, whether the " constant stress " be with p or without p, or with the mean of the two : thus for air p must be a small fraction of the whole pressure, for instance a small fraction of one atmosphere for air at ordinary pressure; for water or watery solutions of salts or other solids, for mercury, for oil, and for other known liquids p may, for all we know, amount to twenty unrtnospheres or one hundred atmospheres without transgressing the limits for which the preceding formula is applicable. When the law of variation of K and e with pressure is known, the differential formula is readily integrated to give the integral amount of the change of temperature produced by greater stress than those for 1 W. Thomson, "Dynamical Theory of Heat" ( 49), Trans. R.S.E., March 1851, aiid " Thermoelastic Properties of Matter," Quarterly Journal of Mathematics, April 1855 (republished Phil. Ala/j. 1877, second half year). 2 lbid., Part vi. 97, 100, Trans. R.S.E., May 1854. Accord ing to the scale there defined on thermo dynamic principles, inde pendently of the properties of any particular substance, t is found, by Joule and Thomson s experiments, to agree very approximately with temperature centigrade, with 274 added. which the differential formula is applicable. For air and other permanent gases Boyle s law of compression and Charles s law of thermal expansion supply the requisite data with considerable accuracy up to twenty or thirty atmospheres. The result is expressed by the formula /IN where k denotes the ratio of the thermal capacity, pressure constant, to the thermal capacity, volume constant, of the gas, a number which thermodynamic theory proves to be approximately constant for all temperatures and densities, for any fluid approximately fulfilling Boyle s and Charles s laws ; P and t the initial pressure and temperature of the gas ; p the sudden addition to the pressure ; and, as before, 6 the elevation of temperature. For the case of p a small fraction of P the formula gives
- -(*-l)f< .... (2.)
It is by an integration of this formula that (1) is obtained. For common air the value of k is very approximately 1 41. Thus if a quantity of air be given at 15 C. (t = 289) and the ordinary atmospheric pressure, and if it be compressed gradually up to 32 atmospheres, or dilated to g^- of an atmosphere, and perfectly guarded against gain or loss of heat from or to without, its temperature at several different pressures, chosen for example, will be according to the following table of excesses of temperature above the primitive temperature, calculated by (1). TABLE SHOWING EFFECTS OF PRESSURE ON TEMPERATURE. Air given at temperature 15 Cent. (289 absolute). Value of P-hp. Elevation of temperature produced by com pression. Value cf i +p. Lowering of temperature produced by dilata tion. 2 95 i 71 4 8 221 389
i 125 166 16 32 612 911 TYT TS "2 196 219 But we have no knowledge of the effect of pressures of several thousand atmospheres in altering the expansibility or specific heat in liquids, or in fluids which at less heavy or at ordinary pressures are " gases." 75. When change of temperature, whether in a solid or a fluid is produced by the application of a stress, the corresponding modulus of elasticity will be greater in virtue of the change of temperature than what may be called the static modulus defined as above, on the understanding that the temperature if changed by the stress is brought back to its primitive degree before the measurement of the strain is performed. The modulus calculated on the supposition that the body, neither losing nor gaining heat during the application of the stress and the measurement of its effect, retains the whole change of temperature due to the stress, will be called for want of a better name the kinetic modulus, because it is this which must (as in Laplace s celebrated correction of Newton s calculation of the velocity of sound) be used in reckoning the elastic forces concerned in waves and vibrations in almost all practical cases. To find the ratio of the kinetic to the static modulus remark that e6, according to the notation of section 74, is the diminution of the strain duo to the change of temperature 0. Hence if M denote the static modulus (section 41), the strain actually produced by it when the body is riot allowed either fp to gain or luse heat is {- - cO, or, with replaced by its 31
value according to the formula of section 74,
