« EelmineJätka »
product of Mr into the moment of inertia of the area round the perpendicular to its plane through its centre, which is therefore equal to the moment of the couple applied at either end.
66. Prism of any inapt constrained to a Simple Twist.— Farther, it is easily proved that if a cylinder or prism of any shape be compelled to take exactly the state of strain above specified (section 65) with the line through the centres of inertia of the normal sections, taken instead of the axis of the cylinder, the mutual action between the parts of it on the two sides' of any normal section will be a couple of which the moment will be expressed by the same formula, that is, the product of the rigidity, into the rate of twist, into the moment of inertia of the section round its centre of inertia. But for any other shape of prism than a solid or symmetrical hollow circular cylinder, the supposed state of strain requires, besides the terminal opposed couples, force parallel to the length of the prism, distributed over the prismatic boundary, in proportion to the distance FE along the tangent, from each point of the surface, to the point in which this line is cut by a perpendicular to it from 0 the centre of inertia of the normal section. . To prove this let a normal section of the prism be represented in the annexed diagram (fig. 6). Let PK, representing the shear at any.point P,'close to the prismatic boundary, be resolved into FN and PT along the normal and tangent respectively. The whole, shear FE being equal to rr its component FN is equal to rr sin u or r.PE The corresponding component of the required stress is nr. FE,
and involves equal forces in , the plane of the diagram, and in the plane through TP perpendicular to it, each amounting to nr. PE per unit of area.
An application of force equal and opposite to the distribution thus found over the prismatic boundary, would of course alone produce in the prism, otherwise free, a state of strain which, compounded with that supposed above, would give the state of strain actually produced by the sole application of balancing couples to the two ends. The result, it is easily seon, consists of an increased twist, together with a warping of naturally plane normal sections, by infinitesimal displacements perpendicular to themselves, into certain surfaces of anticlastic curvature, with equal opposite curvatures. In bringing forward this theory, St Yenant not only pointed out the falsity of the supposition admitted by several previous writers, and used in practice fallaciously by engineers, that Coulomb's law holds for other forms of prism than the solid or hollow circular cylinder, but he discovered fully the nature of the requisite correction, reduced the determination of it to a problem of pure mathematics, worked ont the solution for a great variety of important and curious cases, compared the results with observation in a manner satisfactory and interesting to the naturalist, and gave conclusions of great value to the practical engineer.
67. "Hydrokinetic Analogue to Torsion Problem.*—We take advantage of the identity of mathematical conditions in St Venanf a torsion problem, and a hydrokinetic problem first solved a few years earlier by Stokes,* to give the following statement, which will be found very useful in estimating deficiencies in torsional rigidity below the amount calculated from the fallacious extension of Coulomb's law:—
"Conceive a liquid of density n completely filling a closed infinitely light prismatic box of the same shape within as the given elastic prism and of length unity, and let a couple be applied to the box in a plane perpendicular to its length. The effective' moment of inertia of the liquid* will be equal to the correction by which the torsional rigidity of the elastic prism, calculated by the false extension of Coulomb's law, must be diminished to give the true torsional rigidity.
'.< Farther, the actual shear of the solid, in any infinitely thin plate of it between two normal sections, will at each point be, when reckoned as a differential sliding (section 43) parallel to their planes, equal to and in the same direction as the velocity of the liquid relatively to the containing box."
68. Solution of Torsion Problem.—To prove these propositions and investigate the mathematical equations of the problem, the process followed in Thomson and Tait's Natural Philosophy, section 706, is first to show that the conditions of sections 63 are verified by a state of strain compounded of (1) a simple twist round the line through the centres of inertia, and (2) a distortion of eaih normal section by infinitesimal displacements perpendicular to its plane; then find the interior and surface equations to determine this warping; and lastly, calculate the actual moment of the couple to which the mutual action between the matter on the two sides of anv normal section is equivalent .
69. St Variant's treatise abounds in beautiful and instructive graphical illustrationa of his results, from which the following are selected >
(1.) Elliptic Cylinder.—The plain and dotted cnrvilineal arcs are (fig. 7) "contour lines" (coupes topographiquet) of the section as warped by torsion; that is to say, lines in which it is cut by a series of parallel planes, each perpendicular to the axis. The arrows indicate the direction of rotation
in the part of the prism above the plane of the
(5.) Diagram of St Tenant's curvilineal squares for which torsion problem is algebraically solvable.—This diagram (fig. 11) shows the series of lines represented by the equation x'+y1 -a(x* - Gxty + y*) — 1 -a, with the indicated values for a. It is remarkable that the values a •= 0 5 and a = ~ !( V "" 1) 8'Te similar but not equal curvilineal squares (hollow sides and acute angles), one of them turned through half a right angle relatively to the other.
70. Torsional Rigidity less in proportion to 'um of principal Flexural Rigidities than according to false ertension (section 66) of Coulomb's Law.—Inasmuch as the moment of inertia of a plane area about an axis through its centre of inertia perpendicular to its plane is obviously equal to the sum of its moments of inertia round any two axes through the same point at right angles to one another in its plane, the fallacious extension of Coulomb's law, referred to in section 66, would make the torsional rigidity of a bar of any section equal to the product of the ratio of the modulus of rigidity to the Young's modulus into the sum of its flexural rigidities (section 61) in any two planes at right angles to one another through its length. The true theory, as we have seen (section 67), always gives • torsional rigidity less than this. How great the deficiency
analogy given in section 07), has been pointed ont by M. de St Venant, with the important practical application, that strengthening ribs, or projections (see, for instance, the second of the annexed diagrams), such as are introduced in engineering to give stiffness to beams, have the reverse of a good effect when torsional rigidity or strength is an object, although they are truly of great value in increasing the flexural rigidity, and giving strength to bear ordinary strains, which are always more or less fiexuraL With remarkable ingenuity and mathematical skill he has drawn beautiful illustrations of this important practical principle from bis algebraic and transcendental solutions.
Tig. 12.—Diigranu showing toraiontl rigidities.
Thus, for an equilateral triangle, and for the rectilineal and three curvilineal squares shown in the diagrams (fig. 12), he finds for the torsional rigidities the values stated. The number immediately below the diagram indicates in each case the fraction which the true torsional rigidity is of the old fallacious estimate (section 66),—the latter being the product of the rigidity of the substance into the moment of inertia of the cross section round an axis perpendicular to its plane through its centre of inertia. The second number indicates in each case the fraction which the torsional rigidity is of that of a solid cirsular cylinder of the same sectional area.
71. Places of greatest Distortion in Twisted Prisms.—M. de St Venant also calls attention to a conclusion from his solutions which to many may be startling, that in his eimpler cases the places of greatest distortion are those j. Mnts of the boundary which are nearest to the axis of the twisted prism in each case, and the places of least distortion those farthest from it. Thus in the elliptic cylinder the
fcubstuuce is niobt (trained at vhe euds of the smaller principal diameter, and least at the ends of the greater. In the equilateral triangular and square piisnis 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 aides, 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, excessively 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, as (6) 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 Anglet, infinite, Liability to Cracks proceeding from Reentrant Anglet, or any places of too sharp concave curvature,—A solid of any elastic substance, isotropic or ■eolotropic, 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 neighbouring 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 tc*ion 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).
a> o.) <».) «.i \>> i* J
The solution shows that whnn the solid is continuous from the circular cylindrical surface to its axis, as in (4), (6), (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 theory1 it is concluded that cold is produced whenever a solid is strained .by opposing, 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 condensation or dilatation, uniform in all directions, a fluid may be
* W. Thomson on "Thermo-elastic Properties of Matter," in Quarterly. Journal of Mathematics, April 1865 (republished in Phil. Mag. 1377, second hatf year.) — ~~ ~
included in the statement. Hence the following propositions :—
(1.) A cubical compression of any elastic fluid or solid in an ordinary condition causes an evolution of heat; hut, 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 bo 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 experimentally verified by Joule (" Thermodynamic Properties of Solids," Trans. Roy. Sac., 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 diminution 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 diminished pull lowering of temperature.
(6.) 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 Oough (Memoirs of the Literary and Philosophical Society of Manchester, 2d series, voL i. p. 288), quoted by Joule in his paper on "Thermodynamic Properties of Solids" (Transaetions of Royal Society, 1858). The thermodynamic conclusion 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 thermodynamic 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 indiarubber 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 redhot body is near, and falhi 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 ho hud observed in the same piece of india-rubber.
74 The thermodynamic theory gives one formula1 by which the change of temperature in every such case may be calculated when the other physical properties are known:—
where 6 denotes the elevation of temperature produced by the sadden application of a stress p * t, the temperature of the substance on the absolute thermodynamic scale,* the change of temperature 0 being supposed to be bat 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;
E, 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 most 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 perpendicular to a set of parallel planes, e most 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,« 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. Tb,, 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. Mathematically speaking, p is to be infinifiwima^ bat practically it may be of any magnitude moderate enough not to give any sensible difference in the value of either K or «, 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 atmospheres or one hundred atmospheres without transgressing the limits for which the preceding formula is applicable. When the law of variation of K and « 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), Tram. R.S.E., March 1851, and "Therm oebiatio Propertiea of Matter," Quarterly Journal of Mathematics, April 1855 (republished Phil. Mag. 1877, second half year).
< Ibid., Part vi. §§ 97, 100, Tram. R.S.B., Hay 1864. According to the acal* then defined on ther -10 dynamic principles, independently of the propertiae of any particular substance, t is found, by Joule and Thomson's experiment*, to agree To 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
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 gaa;
p the sudden addition to the pressure;
and, as before, $ the elevation of temperature.
For the case of p a small fraction of F the formula gives
* = (*-l)£f .... (2.)
It is by an integration of this formula that (1) is obtained.
For common air the value of £ is very approximately 141. Thus if a quantity of air be given at 16* C. ((•> 289°) and the ordinary atmospheric pressure, and if it be compressed gradually up to 32 atmospheres, or dilated to To 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).
But we have no knowledge of the effect of pressures of several thousand atmospheres in altering the expansibility or specific hest 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 daring the application of the stress and the measurement of its effect, retains the whole change of temperature doe to the stress, will be called for want of abetter name the kinetic modulus, because it is this which must (ss 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 due to the change of temperature 6. Hence if M denote the static modulus (section 41), the strain actually produced by it when the body is not allowed either
to gain or lose heat is ^ - e6, or, with 0 replaced by its
THERMODYNAMIO TABLE II.
Pressure parallel to one direction in a solid-Ratio of Kinetic to Dividing p by this approasion wo find for the kinetic Static Young's Modulus. Temperature 15° C. (289o absolute). modulus
value of Thermal
pall Capacity, Expan
por anit ability
grammos of ono
grammo per square
metre M te M
-09970000249 .000000308 873 x 10° 1.0080 n.
0314 000029 000000394 76. For any substance, fluid or solid, it is easily proved,
417 x 100 1.00362
Silver .. 10-369 0557 -000019 000000224 736 x 10° 1 00315 without thermodynamic theory, that
Copper . 8-933 0949 000018000000145 1245 x 10° 1 00325
11-215 0293 -000029 *000000609 177 x 10 1-00310 Glan
7949 177 -0000086 000000113 614 4 x 10° 1.000600
Iron . . 7.833 - 1098 -000018000000107 1861 x 10% 1.06259 MÑ
Platinum 21.276 -0914 1.0000086-0000000778 1704 x 10° 1:00199 where K denotes the thermal capacity of a stated quantity of the substance under constant stress, and N its thermal
77. Experimental Results.—The following tables show capacity under constant strain (or thermal capacity when determinations of moduluses of compression, of Young's the body is prevented from change of shape or change of modulus, and of moduluses of rigidity by various experi. volume). For permanent gases, and generally for 'fluids menters and various methods. It will be seen that the approximately fulfilling Boyle's and Charles's laws as said
Young's moduluses obtained by Wertheim by vibrations, above, k is proved by thermodynamic theory to be approxi
longitudinal or transverse, are generally in excess of those mately constant. Its value for all gases for which it bas which he found by static extension; but the differences been measured differs largely from unity, and probably also are enormously greater than those due to the heating and for liquids generally (except water near its temperature of cooling effects of elongation and contraction (section 76), maximum density).
and are to be certainly reckoned as errors of observation. On the other hand, for solids whether the stress con. It is probable that his moduluses determined by statio sidered be uniform compression in all directions or of any elongation are minutely accurate; the discrepancies of those
M' K other type, the value of or differs but very little found by vibrations are probably due to imperfections of the
arrangements for carrying out the vibrational method :) from unity; and both for solids and liquids it is far from constant at different temperatures (in the case of water it is
TABLE OF MODULUSZA OF CUMPRENSIBILITY. zoro at 3o.9 Cent., and varies as the square of the differenco of the temperature from 309 at all events for moderato
Mudulunos differences from this critical temperature, whether above or
yrussibility in TempeSubstance.
MK below it). The following tables show the value of or, and the value of 0 by the formula of sec. 74, for different fluid and solid substances at the temperature 15° Cent.
22.63 x 10 15 (289o absolute scale).
124 x 10° The first table is for compression
Amaury and 11.4 x 108
Descampi, uniform in all directions; the second, necessarily confined
9.5 x 108
Comptes Ren. to solids, is for the stress dealt with in “Young's Modu.
8.07 x 10 11 dus, tome xvii. lus," that is, normal pressure (positive or negative) on one
Bisulphido of carbon 16.3 x 10% ] p. 1564 (1869). Bet of parallel planes, with perfect freedom to expand or
552.5 x 109
Everett's NI:18. contract in all directions in these planes. A wire or rod
trations of the
354 x 10
Another specimen: polled longitudinally is a practical application of the
1876 x 106 latter.
1485 x 108
Second System THERMODYNAMIC TABLE I.
1717 x 108 .
of Units. Pressure equal in all directions-Ratio of Kinetic to Static Bulk
( Wertheim, Brass, different speci
Ann. de Temperature 15° C. (289o absolute) J-42400 centi. mens . .' 1063 x 10
( Chim., 1848. metres.
yer yuero centimetru.