EL-ARAISH, L'Araish, or in French Laracπe, a town of Morocco en the Atlantic coast, about 45 miles S. of Tangier, is picturesquely situated on a rocky height to the south of the embouchure of the Wady Loukhus or Lixus. It is the seat of a military governor, and has a number of well-kept though practically useless defences. The impress of Spanish occupation is still evident, and all the main points described in the 17th century by Pidon de Saint Olon can easily be distinguished—such as the church, the fort of St Jacques, the castle of St Etienne with its four cupolas, the Jew’s Tower, and the castle of Notre Dame d’Europe, now the Kasba or citadel. The market-place is surrounded with arcades of monolithic sandstone pillars. In spite of the bar at the entrance of the river preventing the passage of all vessels of more than 150 tons, the port is one of the most frequented on that part of the coast. The exports, gradually increasing in value, consist mainly of millet, drà, and other cereals, canary-seed, beans, pease, cork, and wool. In 1875, 136 vessels entered and cleared, 26 being British and 58 Spanish. The population of the town at the same date was estimated at 5000, of whom nearly 4000 were Mahometans, about 1000 Spanish-speaking Jews, and 60 Christians.
Though the name of El-Araish is comparatively modern, and is mentioned neither by El-Bikri nor by Edrisi, it seems not improbable from a passage in Scylax that the site of the town was occupied by a Libyan settlement at an early date; and about 3⅛ miles up the river there still exist on the hill of Tchemmish very considerable ruins of the Punico-Roman city of Lixus. The modern town was finally taken from the Portuguese in 1689 by Mulei Ismael after a five months’ siege; in 1785 it was attacked by the French, and in 1829 saw the destruction of the Morocco fleet by the Austrians. A convent in connection with the Spanish mission was maintained till 1822.
See Barth, Wanderungen durch die Küstenländer des Mittelmeeres, 1S49j Rohlf’s Adventures in Morocco, 1874; Tissot, u Itinéraire de Tanger à R’ba-t,” in Bull, de la Soc. de Geogr., 1876.
1. I il LASTICITY of matter is that property in virtue of
J_J which a body requires force to change its bulk or shape, and requires a continued application of the force to maintain the change, and springs back when the force is removed, and, if left at rest without the force, does not remain at rest except in its previous bulk and shape. The elasticity is said to be perfect when the body always requires the same force to keep it at rest in the same bulk and shape and at the same temperature through whatever variations of bulk, shape, and temperature it be brought. A body is said to possess some degree of elasticity if it requires any force to keep it in any particular bulk or shape. It is convenient to discuss elasticity of bulk and elasticity of shape sometimes separately and sometimes jointly.
2. Every body has some degree of elasticity of bulk. If a body possesses any degree of elasticity of shape it is called a solid; if it possesses no degree of elasticity of shape it is called a fluid.
3. All fluids possess elasticity of bulk to perfection. Probably so do all homogeneous solids, such as crystals and glasses. It is not probable that any degree of fluid pressure (or pressure acting equally in all directions) on a piece of common glass, or rock crystal, or of diamond, or on a crystal of bismuth, or of copper, or of lead, or of silver, would make it denser after the pressure is removed, or put it into a condition in which at any particular intermediate pressure it would be denser than it was at that pressure before the application of the extreme pressure. Malleable metals and alloys, on the other hand, may have their densities considerably increased and diminished by hammering and by mere traction. By compression between the dies used iu coining, the density of gold may be raised from 19 ’ 258 to 19 φ 367, and the density of copper from 8 ’ 535 to 8’916; and Mr M'Farlane's experiments quoted below (section 78), show a piece of copper wire decreasing in density from 8 ’ 91 to 8 ’ 835 after successive simple tractions, by which its length was increased from 287 centimetres to 317 centimetres, while its modulus of rigidity decreased from 443 to 426 million grammes per square centimetre. Later experiments, recently made for this article by the same experimenter, have shown augmentation of density from 8 ’ 85 to 8 ’ 95, produced by successive tractions which elongated a piece of copper wire from weighing 16 ’ 4 grammes per metre to weighing 13 - 5 grammes per metre, the wire having been first annealed by heating it to redness in sand, and allowing it to cool slowly. Augmentation of density by traction is a somewhat surprising result, but not altogether so when we consider that the wire had been reduced to an abnormally small density by the previous thermal treatment (the “annealing”). The common explanation of these changes of density in metals, which attributes them to porosity, is probably true; by porosity being understood a porous structure with such vast numbers of the ultimate molecules in the portions of the solid substance between pores or interstices that these portions may be called homogeneous in the sense that a crystal or a liquid can be called homogeneous (compare section 40 below).
4. The elasticity of shape of many solids is not perfect: it is not known whether it is perfect for any. It might be expected to ba perfect for glass and rock crystal and diamond and other hard, brittle, homogeneous substances; but experiment proves that at all events for glass it is not so, and shows on the contrary a notable degree of imperfection in the torsional elasticity of glass fibres. It might be expected that in copper and soft iron and other plastic metals the elasticity of shape would be very imperfect; experiment shows, on the contrary, that in copper, brass, soft iron, steel, platinum, provided the distortion does not exceed a certain limit in each case, elasticity of shape is remarkably perfect, much more perfect than in glass. It is quite probable that even in the softer metals—zinc, tin, lead, cadmium, potassium, sodium, &c.—the elasticity of shape may be as perfect as in the metals mentioned above, but within narrower limits as to degree of distortion. Accurate experiment is utterly wanting, to discover what is the degree of imperfection, if any, of the elasticity of any metal or alloy, when tested within sufficiently narrow limits of distortion.
5. The “viscosity of metals” described below (sections 21- 25) does not demonstrate any imperfectness of elasticity according to the definition of section 1, which is purely statical. The viscosity of solids may (for all we yet know by experiment) depend, as does the viscosity of fluids, upon a resistance varying with the velocity of the change, and vanishing when the velocity of the change is zero, that is to say, when the body is at rest in any configuration; if so, the elasticity of the substance concerned is perfect within the limits of the experiment in question. If, on the other hand (as the discovery of elastic fatigue described below seems to indicate may be to some degree the case), the loss of energy from the vibrations in the experiments described is due to a dependence of the elastic resilient force upon previous conditions of the substance in respect to strain, the “viscosity” would be continuous with a true imperfectness of static elasticity. Here, then, we have a definite question which can be answered by experiment only :—Consider a certain definite stress applied to a solid substance; as, for example, a certain “couple” twisting a wire or rod; or a certain weight pulling it out, or compressing it lengthwise; or a certain weight placed on the middle of a beam supported by trestles under its ends. Let it be applied and removed a great many times, and suppose it to be seen that after each application and removal of the stress the body comes to rest in exactly the same configuration as after the previous application or removal of the stress. If now the body be left to itself with the stress removed, and if it be found to remain at rest in the same configuration for minutes, or hours, or days, or years after the removal of the stres3, a part of the definition of perfect elasticity is fulfilled. Or, again, if the stress be applied, and kept applied with absolute constancy, and if the body remain permanently in a constant configuration, another item of the definition of perfect elasticity is proved. When any such experiment is made on any metal, unless some of the softer metals (section 4) is to be excepted, there is certainly very little if any change of configuration in the circumstances now supposed. The writer believes, indeed, that nothing of the kind has hitherto been discovered by experiment, provided the stress has been considerably less than that which would break or give a notable permanent twist, or elongation, or bend, to the body, that is to say, provided the action has been kept decidedly within the limits of the body’s elasticity as commonly understood (sections 7-20). Mr J. T. Bottomley, with the assistance of a grant of money from thθ British Association, has commenced making arrangements for secular experiments on the elasticity of metals, in the tower of the university of Glasgow, to answer this question in respect to permanence or non-permanence through minutes, or hours, or days, or years, or centuries. If several gold wires are hung side by side, one of them bearing the smallest weight that will keep it approximately straight, another wire - 5 1 0 of the breaking weight, another wire - 5 2 σ of the breaking weight, and so on; the one of them bearing ⅜4 of the breaking weight will probably, in the course of a few hours or days, show very sensible elongation. Will it go on becoming longer and longer till it breaks, or will the time-curve of its elongation be asymptotic? Even with considerably less than ∙j-⅜ of the breaking weight there will probably be a continually augmenting elongation, but with asymptotic time-curve indicating a limit beyond which the elongation never goes, but which it infinitely nearly reaches in an infinite time. It is not probable that a gold wire stretched by 1 1 σ of its present breaking weight, or by £ of its present breaking weight, or even by ⅜ of its present breaking weight, would break in a thousand or in a million years. The existence of gold ornaments which have been fouud in ancient tombs and cities, and have preserved their shapes for thousands of years without running down glacier-wise (as does brittle pitch or sealing-wax in the course of a few years in moderately warm climates), seems to prove that for gold (and therefore leaves no doubt also for many other metals) the time-curve is asymptotic, if indeed there is any slow change of shape at all after the application of a moderate stress well within the limits of elasticity. Egyptian and Greek statues, Etruscan vases, Egyptian obelisks, and other stone monuments with their engraved hieroglyphics, flint implements and boulders, and mountains with the geological evidence we have of their antiquity, prove for stones, and pottery, and rocks of various kinds, a permanence for thousands and millions of years of resistance to distorting stress.
6. The complete fulfilment of the definition of perfect elasticity is not proved by mere permanence of the extreme configurations assumed by the substance when a stated amount of the stress is alternately applied and removed. This condition might be fulfilled, and yet the amount of elastic force might be different with the same palpable configuration of the body during gradual augmentation and during gradual diminution of the stress. That it is so in fact is proved by the discovery of viscosity referred to below; but it is not yet proved that if, after increasing the stress to a certain definite amount, the body is brought to rest in the same palpable configuration as before, the amounts of stress required to hold it in this configuration are different in the two cases. If they are (section 1) the elasticity is imperfect; if they are not the elasticity is perfect within the limits of the experiment (compare section 36 below).
7. Limits of Elasticity— Elasticity of Shape.— The degree of distortion within which elasticity of shape is found is essentially limited in every solid. Within sufficiently narrow limits of distortion every solid shows elasticity of shape to some degree—some solids to perfection, so far as we know at present. When the distortion is too great, the body either breaks or receives a permanent bend (that is, such a molecular disturbance that it does not return to its original figure when the bending force is removed). If the first notable dereliction from perfectness of elasticity is a breakage, the body is called brittle,—if a permanent bend, plastic or malleable or ductile. The metals are generally ductile; some metals and metallic alloys and compounds of metals with small proportions of other substances, are brittle; some of them brittle only in certain states of temper, others it seems essentially brittle. The steel of before the days of Bessemer and Siemens is a remarkable instance. When slowly cooled from a bright red heat, it is remarkably tough and ductile. When heated to redness and cooled suddenly by being plunged in oil or water or mercury, it becomes exceedingly brittle and hard (glass-hard, as it is called), and to ordinary observation seems incapable of taking a permanent bend (though probably careful observation would prove it not quite so). The definition of steel used to be approximately pure iron capable of being tempered glass-hard, and again softened to different degrees by different degrees of heat. Now, the excellent qualities of iron made by Bessemer’s and Siemens’s processes are called steel, and are reckoned best when incapable of being tempered glass-hard, the possibility of brittleness supervening in the course of any treatment which the metal may meet with in its manufacture being an objection against the use of what was formerly called steel for ship’s plates, ribs, stringers, &c., and for many applications of land engineering, even if the material could be had in sufficient abundance.
8. Limits of Elasticity (continued)— Elasticity of Bulk.— If we reckon by the amount of pressure, there is probably no limit to the elasticity of bulk in the direction of increase of pressure for any solid or fluid; but whether continued augmentation produces continued diminution of bulk towards zero without limit, or whether for any or every solid or fluid there is a limit towards which it may be reduced in bulk, but smaller than which no degree of pressure, however great, can condense it, is a question which cannot be answered in the present state of science. Would any pressure, however tremendous, give to gold a density greater than 19*6, or to copper a density greater than 9Ό, after the pressure is removed (section 3 above)? But whether the body be fluid or a continuous non-porous solid, it probably recovers the same density, however tremendously it may have been pressed, and probably shows perfect elasticity of bulk (section 3 above) through the whole range of positive pressure from zero to infinity, provided the pressure has been equal in all directions like fluid pressure. As for negative pressure, we have no knowledge of what limit, if any, there may be to the amount of force which can be applied to a body pulling its surface out equally in all directions. The question of how to apply the negative pressure is inextricably involved with that of the body’s power to resist, The upper part of the mercury of a barometer adhering to the glass above the level corresponding to the atmospheric pressure is a familiar example of what is called negative pressure in liquids. Water and other transparent liquids show similar phenomena, another of which is the warming of water above its boiling point in an open glass or metal vessel varnished w ith shellac. Attempts to produce great degrees of this so-called negative pressure are baffled by what seems an instability of the equilibrium which supervenes wflιen the negative pressure is too much augmented. It is a very interesting subject for experimental inquiry to find how high mercury or water or any other liquid can be got to stand above the level corresponding to the atmospheric pressure in a tall hermetically sealed tube, and how many degrees a liquid can, with all precautions, be warmed above its boiling point. In each case ib⅛eems to be by a minute bubble forming and expanding somewhere at the boundary of the liquid, where it is in contact with the containing vessel, that the possible range of the negative pressure is limited, judging from what we see when we carefully examine a transparent liquid, or the surface of separation between mercury and glass, in any such experiment. The contrast of the amounts of negative pressure practically obtainable or obtained hitherto in such experiments on liquids (which are at the most those corresponding to the weight of a few metres of the substance), with that obtainable in the case of even the weakest solids, is remarkable; and as for the strongest, consider for instance (sec. 22 below) 17 nautical miles of steel pianoforte wire hanging by one end. When a cord, or rod, or ware of any solid substance hangs vertically, the negative pressure (for example, 23,000 atmospheres in the case just cited) in any transverse section is equal to the weight of the part hanging below it. It is an interesting question not to be answered by any experiment easily made or even devised,—How much would the longitudinal pull which can be applied to a cord, rod, or wire without breaking it be augmented (probably augmented, but possibly diminished) by lateral pull applied all round the sides so as to give equal negative pressure in all directions?
9. Limits of Elasticity (continued)— Elasticity of Shape for Distortions not Uniform through the Substance, and for Compound Distortions; and Elasticity corresponding to Co-existent Distortion and Change of Bulk :— Example 1.—A round wire twisted, or a cylindrical shaft transmitting revolutional motive in machinery, presents, as we shall see (sec. 64), an instance of simple distortion, but to different degrees in different parts of the substance, increasing from the axis where it is zero, uniformly to the surface where it is greatest.
Example 2.—Elongation of a wire or rod by direct pull, is (sec. 23) an instance of a compound distortion co-existing with a rarefaction of the substance, both distortion and rarefaction uniform throughout.
Example 3.—Shortening of a column by end pressure is an instance of a similar compound distortion combined with condensation of the substance, both distortion and condensation uniform throughout.
Example 4.—Flexure of a round wire or of a bar, or beam, or girder, of any shape of normal section, by opposite bending couples applied at the two ends, is an instance in which one-half of the sιtb-stance is stretched, and the other half shortened with exactly the same combination of distortions and changes of bulk as in examples 2 and 3. The strain More nearly what is now called stress than what is now called strain is meant by “strain " in this article, which was written before Rankine’s introduction of the word stress, and distinct definition of the word strain (see chap. i. of Mathematical Theory below). ] is uniform along the length of the bar, but varies in the cross section in simple proportion to distance from a certain line (sec. 62) through the centre of gravity of the sectional area, which, in the case of a round bar, is the diameter perpendicular to the plane of curvature.
The limits of elasticity in the cases of these four examples are subjects of vital importance in practical mechanics, and a vast amount of careful and accurate observation and experiment, which has given much valuable practical information regarding them, has been gone through by engineers, in their necessary dealings with questions regarding strength of materials. Still there is great want of definite scientific information on the subject of limits of elasticity generally, and particularly on many elementary questions (section 21 below), which force themselves upon us when we endeavour to analyze the molecular actions concerned in such cases as the four examples now before us. Some principles of much importance for guidance in practical as well as theoretical deductions from observations and experiments on this subject were set forth twenty-nine years ago by Professor James Thomson, in an article published in the Cambridge and Dublin Mathematical Journal for November 1848. Nothing is to be gained either in clearness or brevity by any other way of dealing with it than reproducing it in extenso. It is accordingly given here, with a few changes made in it with its author’s concurrence.
It constitutes the following sections, 10-20.
“On the strength of materials, as influenced by the existence or nonexistence of certain mutual strains 1 among the particles composing them. By James Thomson, Μ.A., College, Glasgow.
10. “My principal object in the following paper is to show that the absolute strength of any material composed of a substance possessing ductility (and few substances, if any, are entirely devoid of this property) may vary to a great extent, according to the state of tension or relaxation in which the particles have been made to exist when the material as a whole is subject to no external strain.
11. “Let, for instance, a round bar of malleable iron, or a piece of iron wire, be made red hot, and then be allowed to cool. I ta particles may now be regarded as being all completely relaxed. Let, next, one end of the bar be fixed, and the other be made to revolve by torsion, till the particles at the circumference of the bar are strained to the utmost extent of which they can admit, without undergoing a permanent alteration in their mutual connexion. In this condition, equal elements of the cross section or the bar afford resistances proportional to the distances of the elements from the centre of the bar; since the particles are displaced from their positions of relaxation through spaces which are proportional to the distances of the particles from the centre. The couple which the bar now resists, and which is equal to the sum of the couples due to the resistances of all the elements of the section, is that which is commonly assumed as the measure of the torsional strength of the bar. For future reference, this couple may be denoted by L, and the angle through which it has twisted the loose end of the bar by Θ.
12. “The twisting of the bar may, however, be carried still farther, and during the progress of this process the outer particles will yield in virtue of their ductility, those towards the interior successively reaching their elastic limits, until, when the twisting has been sufficiently continued, all the particles in the section, except those quite close to the centre, have been strained beyond their elastic limits. Hence, if we suppose This supposition may be true for some solids; it is certainly not true for solids generally. A piece of copper or of iron taken in a soft and unstrained condition certainly becomes “harder” when strained beyond its first limits of elasticity, that is to say, its limits of elasticity become wider; and a similar result will probably be found in ductile metals generally. Thus the resistance of the outer elements will be greater than those of tire inner elements in the case described in the text, until the torsion has been pushed so far as to bring about the greatest hardness in all the elements at any considerable distance from the axis. It may be that before this condition has been attained the hardening of the outer elements will have been overdone, and they m ιy have tegun to lose strength, and to have become friable and fissured. The principle set forth in the text is not, however, vitiated by the Incorrectness of a supposition introduced merely for the sake of numerical illustration. ] that no change in the hardness of the substance composing the material has resulted from the sliding of its particles past one another, and that therefore all small elements of the section of the bar afford the same resistance, no matter what their distances from the centre may be, it is easy to prove that the total torsional resistance of the bar is ⅛ of what it was in the former case; or, according to the notation already adopted, it is now ⅜ L.
13. “If, after this, all external stress be removed from the bar, it will assume a position of equilibrium, in which the outer particles will be strained in the direction opposite to that in which it was twisted, and the inner ones in the same direction as that of the twisting, the two sets of opposite couples thus produced among the particles of the bar balancing one another. It is easy to show that the line of separation between the particles strained in one direction and those in the other is a circle whose radius is I of the radius of the bar. The particles in this line are evidently subject to no strain. “This, however, does not fall to be considered in the investigation of the text.” ] when no external couple is applied. The bar with its new molecular arrangement may now be subjected, as often as we please, to the couple ⅝ L without undergoing any farther alteration. Its strength to resist torsion, in the direction of the couple L has therefore been considerably increased. Its strength to resist torsion in the opposite direction has, however, by the same process, been much diminished; for as soor as its free extremity has been made to revolve backwards through an angle This assumes that the limits of elasticity in a substance which has already been strained beyond its limits of elasticity are equal on the two sides of the shape which it has when in equilibrium without disturbing force—a supposition which may be true or may not be true. Experiment is urgently needed to test it; for its truth or falseness is a matter of much importance in the theory of elasticity. ] of ¾ Θ from the position of equilibrium, the particles of the circumference will have suffered the utmost distortion of which they can admit without undergoing permanent alteration. Now, it is easy to prove that the couple required to produce a certain angle of torsion is the same in the new state of the bar as in the old. Hence the ultimate strength of the bar when twisted backwards is represented by a couple amounting to only ⅞ L. But, as we have seen, it is ⅜ L when the wire is twisted forwards. That is, then, The wire in its new state has twice as much strength to resist torsion in one direction as it has to resist it the other.
14 “Principles quite similar to the foregoing, are applicable in regard to beams subjected to cross strain. As, however, my chief object at present is to point out the existence of such principles, to indicate the mode in which they are to be applied, and to show their great practical importance in the determination of the strength of materials, I need not enter fully into their application in the case of cross strain. The investigation in this case closely resembles that in the case of torsion, but is more complicated on account of the different ultimate resistances afforded by any material to tension and to compression, and on account of the numerous varieties in the form of section of beams which for different purposes it is found advisable to adopt. I shall therefore merely make a few remarks on this subject.
15. “If a bent bar of wrought iron or other ductile material be straightened, its particleswill thus be put into such a state that its strength to resist cross strain, in the direction towards which it has been straightened, will be veιy much greater than its strength to resist it in the opposite direction, each of these two resistances being entirely different from that which the same bar would afford were its particles all relaxed when the entire bar is free from external strain. The actual ratios of these various resistances depend on the comparative ultimate resistances afforded by the substance to compression and extension, and also, in a very material degree, on the form of the section of the bar. I may, however, state that in general the variations in the strength of a bar to resist cross strain, which are occasioned by variations in its molecular arrangement, are much greater even than those which have already been pointed out as occurring in the strength of bars subjected to torsion.
16. “What has already been stated is quite sufficient to account for many very discordant and perplexing results which have been arrived at by different experimenters on the strength of materials. It scarcely ever occurs that a material is presented to us, either for experiment or for application to a practical use, in which the particles are free from great mutual strains. Processes have already been pointed out by which we may at pleasure produce certain peculiar strains of this kind. These, or other processes producing somewhat similar strains, are used in the manufacture of almost all materials. Thus, for instance, when malleable iron has received its final conformation by the process termed cold swaging, that is, by hammering it till it is cold, the outer particles exist in a state of extreme compression, and the internal ones in a state of extreme tension. The same seems to be the case in cast iron when it is taken from the mould in which it has been cast. The outer portions have cooled first, and have therefore contracted, while the inner ones still continued expanded by heat. The inner ones then contract as they subsequently cool, and thus they, as it were, pull the outer ones together. That is, in the end the outer ones are in a state of compression and the inner ones in the opposite condition.
17. “The foregoing principles may serve to explain the true cause of an important fact observed by Mr Eaton Hodgkinson in his valuable researches in regard to the strength of cast iron (Report of the British Association for 1837, p. 362). He found, that, contrary to what had been previously supposed, a strain, however small in comparison to that which would occasion rupture, was sufficient to produce a set, or permanent change of form, in the beams on which he experimented. Now this is just what should be expected in accordance with the principles which I have brought forward; for if, for some of the causes already pointed out, various !>arts of a beam previously to the application of an external force ιave been strained to the utmost, when, by the application of such force, however small, they are still farther displaced from their positions of relaxation, they must necessarily undergo a permanent alteration in their connexion with one another, an alteration permitted by the ductility of the material; or, in other words, the beam as a whole must take a set.
18. “In accordance with this explanation of the fact observed by Mr Hodgkinson, I do not think we are to conclude with him that ‘the maxim of loading bodies within the elastic limit has no foundation in nature.’ It appears to me that the defect of elasticity, which he has shown to occur even with very slight strains, exists only when the strain is applied for the first time; or, in other words, that if a beam has already been subjected to a considerable strain, it may again be subjected to any smaller strain in the same direction without its taking a set. It will readily be seen, however, from Mr Hodgkinson’s experiments, that the term ‘ elastic limit,’ as commonly employed, is entirely vague, and must tend to lead to erroneous results.
19. “The considerations adduced seem to me to show clearly that there really exist two elastic limits for any material, between which the displacements or deflexions, or what may in general be termed the changes of form, must be confined, if we wish to avoid giving the material a set, or, in the case of variable strains, if we wish to avoid giving it a continuous succession of sets which would gradually bring about its destruction; that these two elastic limits are usually situated one on the one side and the other on the opposite side of the position which the material assumes when subject to no external strain, though they may be both on the same side of this position of relaxation; and that they may therefore with propriety be called the superior and the inferior limit of the change of form of the material for the particular arrangement which has been given to its particles; that these two limits are not fixed for any given material, but that, if the change of form be continued beyond either limit, two new limits will, by means of an alteration in the arrangement of the particles of the material, be f;iven to it in place of those which it previously possessed; and astly, that the· processes employed in the manufacture of materials are usually such as to place the two limits in close contiguity with one another, thus causing the material to take in the first instance a set from any strain, however slight, while the interval which may afterwards exist between the two limits, alιd also, as was before stated, the actual position assumed by each of them are determined by the peculiar strains which are subsequently applied to the material.
20. “The introduction of new, though necessary, elements into the consideration of the strength of materials may, on the one hand, seem annoying from rendering the investigations more complicated. On the other hand, their introduction will really have the effect of obviating difficulties, by removing erroneous modes of viewing the subject, and preventing contradictory or incongruous results from being obtained by theory and experiment. In all investigations, in fact, in which we desire to attain or to approach nearly to truth, we must take facts as they actually are, not as we might be tempted to wish them to be for enabling us to dispense with examining processes which are somewhat concealed and intricate but are not the less influential from their hidden character.”
21. Passing now to homogeneous matter (sec. 38), homogeneously strained (chap. ii. of Math. Theory below), we are met by physical questions of great interest regarding limits of elasticity. Supposing the solid to be homogeneously distorted in any particular way to nearly the limit of its elasticity for this kind of distortion, will the limits be widened or narrowed by the superposition of negative or positive pressure equal in all directions producing a dilatation or a condensation? It seems probable that a dilatation would narrow the limits of elasticity, and a condensation widen them. This, however, is a mere guess: experiment alone can answer the question. Take again a somewhat less simple case. A wire is stretched by a weight to nearly its limits of longitudinal elasticity; a couple twisting it is applied to its lower end—Will this either cause the weight to run down and give the wire a permanent set, or break it? Probably,—yes; but experiment only can decide. The corresponding question with reference to a column loaded with a weight may have the same answer, but not necessarily so. Experiment again is wanting. A wire hanging stretched by a light weight, merely to steady it, is twisted to nearly its limit of torsional elasticity by a couple of given magnitude applied to its lower end ∙ the stretching weight is increased—Will this cause it to yield to the couple and take a permanent set? Probably,— yes. [Certainly yes, for steel piano-forte wire experimented on by Mr M i Farlane to answer this question since it was first put in type fur the present article.] If so, then the limits of torsional elasticity of a wire bearing a heavy weight are widened by diminishing or taking off the weight; and no doubt it will follow continuously that a column twisted by opposing couples at its two ends will have its limits of torsional elasticity widened by tho application of forces to its two ends, pressing them towards one another. Experiments to answer these questions would certainly reward the experimenter with definite and interesting results.
22. Narrowness of Limits of Elasticity— Solids.— The limit of elasticity of metals, stones, crystals, woods, are so narrow that the distance between any two neighbouring points of the substance never alters by more than a small proportion of its own amount without the substance either breaking or experiencing a permanent set, and therefore the angle between two lines meeting in any point of the substance and passing always through the same matter is never altered by more than a small fraction of the radian, before the body either breaks or takes a permanent set. By far the widest limits of elasticity hitherto discovered by experiment, for any substance except cork, india-rubber, jellies, are those of steel pianoforte wire. Take, for example, the piano-forte wire at present in use for deep-sea soundings. It is No. 22 of the Birmingham wire gauge, its density is 7 - 727, it weighs 0 ’ 34 gramme per centimetre, or G ’ 298 kilogrammes per nautical mile of 1852 ’ 3 metres, and therefore its sectional area and diameter are Ό044 square centimetre and Ό244 centimetre. It bears a weight of 106 kilogrammes, which is equal in weight to about 31 kilometres of its length, and when this weight is alternately hung on and removed the length of the wire varies by ∙ g ⅛ of its amount. While this elongation takes place there is a lateral shrinking, as we shall see (section 47), of from ∣ to -j⅜∙ of the same amount.
23. Consider now in the unstrained wire two lines through the substance of the wire at right angles to one another in any plane through or parallel to the axis of the wire in directions equally inclined to this line. When the wire is pulled lengthwise the two vertical angles bisected by the length of the wire become acute, and the other two obtuse by a small difference, as illustrated in the diagram (fig. 2), where the continuous lines represent a portion of the unpulled wire, and the dotted lines the same portion of the wire when pulled. The change m each of the angles would be - g ⅛- of the radian in virtue of the elongation were there no lateral shrinking, and about π ⅜- σ of the radian in virtue of the lateral shrinking were there no elongation. Thewhole change experienced by each of the right angles is therefore actually (section 37) κV ÷ τ⅜σ> or a bout ∙ δ ⅛ of the radian, or 0* ’ 84. This is an extreme case. In all other cases of metals, stones, glasses, crystals, the substance either breaks or takes a permanent bend, probably before it experiences any so great angular distortion as a degree; and except in the case of steel we may roughly regard the limits of elasticity as being something between τ θ⅛θ anc ^ τ⅛υ * n rθspθct to the linear elongation or contraction, and from J σ of a degree to half a degree in respect to angular distortion.
24. On the other hand, gelatinous substances, such as india-rubber and elastic jellies, have very wide limits of elasticity. A vulcanized india-rubber band, for instance, is capable of being stretched, again and again, to eight times its length, and returning always to nearly its previous condition when the stress is removed. A shape of transparent jelly presents a beautiful instance of great degrees of distortion with seemingly very perfect elasticity. All these instances, india-rubber and jellies, show with great changes of shape but slight changes of bulk. They have, in fact, all, as nearly as experiment has hitherto been able to determine, the same compressibility as water.
25. Cork, another body with very wide limits of elasticity (very imperfect elasticity it is true) is singular, among bodies seemingly homogeneous to the eye, in its remarkably easy compressibility. It is, in fact, the only seemingly homogeneous solid which shows to the unaided eye any sensible change of bulk under any practically applicable forces. A small homogeneous piece torn out of a cork may, by merely pressing it between the fingers, be readily compressed to half its bulk, and a large slab of cork in a Bramah press may be compressed to 1 1 σ ∙ of its bulk. An ordinary bottle cork loaded with a small piece of metal presents a very interesting appearance in an Oersted glass compressing vessel; first floating, and when compressed to 20 or 30 atmospheres sinking, and shrivelling in bulk very curiously; then on the pressure being removed, expanding again, but not quite to previous bulk, and floating up or remaining down according to the amount of its load.
The divergencies presented by cork and gelatinous bodies in opposite directions from the regular elasticity of hard solids form an interesting subject, to which we shall return later (section 48).
26. Liquids.— In respect to liquids, there are no limits of elasticity so far as regards the magnitude of the positive pressure applied or conceivably applicable; but in respect to the magnitude of negative pressure, and in respect to the magnitude of the change of bulk, whether by negative or positive pressure, there are probably very decided and not very wide limits. Thus water, though condensed 0 f its bulk by 2000 atmospheres in Perkins’s experiments corrected roughly for the compres sion of his glass “piezometer,” which is very nearly at the rate of 2 - 1 ⅞ 0 ∙ g per atmosphere found (section 75 below) more accurately by subsequent experiments for moderate pressures up to 20 or 30 atmospheres, may be expected to be compressed by much less than ⅜ of its volume under a pressure of 7000 atmospheres. How much it or any other liquid is condensed by a pressure of 10,000 atmospheres, or by 20,000 atmospheres, is an interesting subject for experimental investigation.
27. Gases. —In respect to rarefaction, and in respect to proportionate condensation, gases present enormously wider limits of elasticity than any liquids or solids,—in fact no limit in respect to dilatation, and in respect to condensation adefinite limit only when the gas is below Andrews’s “critical temperature.” If the gas be kept at any temperature above that critical temperature, it remains homogeneous, however much it be condensed; and therefore for a fluid above the critical temperature there is, in respect to magnitude of pressure, no superior limit to its elasticity. On the other hand, if a fluid be kept at any constant temperature less than its critical temperature, it remains homogeneous, and presents an increasing pressure until a certain density is reached; when its bulk is further diminished it divides into two parts of less and greater density (the part of less density being called vapour, that of greater density being called liquid, if it is not solid) and presents no further increase of pressure until the vaporous part shrinks to nothing, and the whole becomes liquid (that is to say, homo geneous fluid at the greater of the two densities) or else becomes solid—the question whether the more dense part is liquid or solid depending on the particular temperature below the critical temperature at which the whole substance is kept during the supposed experiment.
28. The thermo-dynamic reasoning of Professor James Thomson, which showed the effect of change of pressure in altering the freezing point of a liquid, leads to analogous considerations regarding the effect of continuous increase or continuous decrease of pressure upon a mass consisting of the same substance partly in the liquid and partly in the solid state at one temperature. The three cases of transition from gas to liquid, from gas to solid, and from liquid to solid, present us with perfectly definite limits of elasticity, —the only perfectly definite limits of elasticity in nature of which we have any certain knowledge.
29. Viscosity of Fluids and Solids.— Closely connected with limits of elasticity, and with imperfectness of elasticity, is viscosity, that is to say, resistance to change of shape depending on the velocity of the change. The full discovery of the viscosity of liquids and gases is due originally to Stokes; and his hypothesis that in fluids the force of resistance is in simple proportion to the velocity of change of shape has been subsequently confirmed by the experimental investigations of Helmholtz, Maxwell, Meyer, Kundt, and Warburg. The definition of a fluid given in section 2 above may, by section 1, be transformed into the following:—A fluid is a body which requires no force to keep it in any particular shape, or—A fluid is a body which exercises no permanent resistance to a change of shape. The resistance to a change of shape presented by a fluid, evanescent as it is when the shape is not being changed (or vanishing when the velocity of the change vanishes), is essentially different from that permanent resistance to change of shape, the manifestation of which in solids constitutes elasticity of shape as defined in section 1. Maxwell’s admirable kiretic theory of the viscosity of gases points to a full explanation of viscosity, whether of gases, liquids, or solids, in the consideration of configurations and arrangements of relative motions of molecules, permanent in a solid under distorting stress, and temporary in fluids or solids while the shape is being changed, in virtue of which elastic force in the quiescent solid, and viscous resistance to change of shape in the non-quiescent fluid or solid, are produced.
30. Viscosity of Metals and Fatigue of their Elasticity.— Experimental exercises performed by students in the physical laboratory of the university of Glasgow, during the session 1864-65, brought to light some very remarkable and interesting results, proving a loss of energy in elastic vibrators (sometimes as much as two or three per cent, of energy lost in the course of a single vibration in one direction) incomparably greater than anything that could be due to imperfections in their elasticity (section 1), and showing also a very remarkable fatigue of elasticity, according to which a wire which had been kept vibrating for several hours or days through a certain range came to rest much quicker when left to itself than when set in vibration after it had been at rest for several days and then immediately left to itself. Thus it was found that the rates of subsidence of the vibrations of the several wires experimented on were generally much less rapid on the Monday mornings, when they had been at rest since the previous Friday, than on other days of the week, or than after several series of experiments had been made on a Monday. The following statement (sections 31-34) is extracted from a short article by W. Thomson, in the Proceedings of the Royal Society for May 18, 1865, containing some of the results of these observations.
31. “Viscosity.— By induction from a great variety of observed phenomena, we are compelled to conclude that no change of volume or of shape can be produced in any kind of matter without dissipation of energy. Even in dealing with the absolutely perfect elasticity of volume presented by every fluid, and possibly by some solids, as for instance homogeneous crystals, dissipation of energy is an inevitable result of every change of volume, because of the accompanying change of temperature, and consequent dissipation of heat by conduction or radiation. The same cause gives rise necessarily to some degree of dissipation in connection with every change of shape of an elastic solid. But estimates founded on the thermodynamic theory of elastic solids, which I have given elsewhere, have sufficed to prove that the loss of energy due to this cause is small in comparison with the whole loss of energy observed in many cases of vibration. I have also found, by vibrating a spring alternately in air of ordinary pressure and in the exhausted receiver of an air-pump, that there is an internal resistance to its motions immensely greater than the resistance of the air. The same conclusion is to be drawn from the observation made by Kupffer in his great work on the elasticity of metals, that his vibrating springs subsided much more rapidly in their vibrations than rigid pendulums supported on knife-edges. The subsidence of vibrations is probably more rapid in glass than in some of the most elastic metals, as copper, iron, silver, aluminium; but it is much more rapid than in glass, marvellously rapid indeed, in some metals (as for instance zinc), and in india-rubber, and even in homogeneous jellies.
32. “The frictional resistance against change of shape must in every solid be infinitely small when the change of shape is made at an infinitely slow rate, since, if it were finite for an infinitely slow change of shape, there would be infinite rigidity, which we may be sure does not exist in nature. Hence there is in elastic solids a molecular friction which may be properly called viscosity of solids, because, as being an internal resistance to change of shape depending on the rapidity of the change, it must be classed with fluid molecular friction, which by general consent is called viscosity of fluids. But, at the same time, it ought to be remarked that the word viscosity, as used hitherto by the best writers, when solids or heterogeneous semi-solid semifluid masses are referred to, has not been distinctly applied to molecular friction, especially not to the molecular friction of a highly elastic solid within its limits of high elasticity, but has rather been employed to designate a property of slow continual yielding through very great, or altogether unlimited, extent of change of shape, under the action of continued stress. It is. in this sense that Forbes, for instance, has used the word in stating that ‘ viscous theory of glacial motion,’ which he demonstrated by his grand observations on glaciers. As, however, he and many other writers after him have used the words plasticity and plastic, both with reference to homogeneous solids (such as wax or pitch even though also brittle, soft metals, &c.) and to heterogeneous semi-solid semi-fluid masses (as mud, moist earth, mortar, glacial ice, &c.), to designate the property common to all those cases of experiencing, under continued stress, either quite continued and unlimited change of shape, or gradually very great change at a diminishing (asymptotic) rate through infinite time, and as the use of the term plasticity implies no more than does viscosity any physical theory or explanation of the property, the word viscosity is without inconvenience left available for the definition I propose.
33. “To investigate the viscosity of metals, I have in the first place taken them in the form of round wires, and have chosen torsional vibrations, after the manner of Coulomb, for observation, as being much the easiest way to arrive at definite results. In every case one end of the wire was attached to a rigid vibrator with sufficient firmness (thorough and smooth soldering I find to be always the best plan when the wire is thick enough); and the other to a fixed rigid body, from which the wire hangs, bearing the vibrator at its lower end. I arranged sets of observations to be made for the separate comparison of the following cases :—
(a) “The same wire with different vibrators of equal weights to give equal stretching-tractions but different moments of inertia (to test the relation between viscous resistances against motions with different velocities through the same range and under the same stress).
(ό) “The same wire with different vibrators of equal moments of inertia but unequal weights (to test the effect of different longitudinal tractions on the viscous resistance to torsion under circumstances similar in all other respects).
(c) “The same wire and the same vibrator, but different initial ranges in successive experiments (to test an effect unexpectedly discovered, by which the subsidence of vibrations from any amplitude takes place at very different rates according to the immediately previous molecular condition, whether of quiescence or of recurring changes of shape through a wider range).
(<Z) “Two equal and similar wires, with equal and similar vibrators, one of them kept as continually as possible in a state of vibration, from day to day; the other kept at rest, except when vibrated in an experiment once a day (to test the effect of continued vibration on the viscosity of a metal).
34. “Results.—(a) It was found that the loss of energy in a single vibration through one range was greater the greater the velocity (within the limits of the experiments); but the difference between the losses at low and high speeds was much less than it would have been had the resistance been, as Stokes has proved it to be, in fluid friction, approximately as the rapidity of the change of shape. The irregularities in the results of the experiments which up to this time I have made seem to prove that much smaller vibrations (producing less absolute amounts of distortion in the parts of the wires most stressed) must be observed before any simple law of relation between molecular friction and velocity can be discovered.
(δ) “When the weight was increased, the viscosity was always at first much increased; but then day after day it gradually diminished and became as small in amount as it bad been with the lighter weight. It has not yet been practicable to continue the experiments long enough in any case to find the limit to this variation.
(c) “The vibration subsided in aluminium wires much more rapidly from amplitude 20 to amplitude 10, when the initial amplitude was 40, than when it was 20. Thus, with a certain aluminium wire, and vibrator No. 1 (time of vibration one way 1 ’ 757 second), the number of vibrations counted were in three trials— [table]
(<7) “In a wire which was kept vibrating nearly all day, from day to day, after several days very much more molecular friction was found than in another kept quiescent except during each experiment. Thus two equal and similar pieces of copper wire were put up about the 26th of April, hanging with equal and similar lead weights, the upper and lower ends of the two wires being similarly fixed by soldering. No. 2 was more frequently vibrated than No. 1 for a few days at first, but no comparison of viscosities was made till May 15. Then [table]
During the greater part of May 16 and 17, No. 2 was kept vibrating and No. 1 quiescent, and late on May 17 experiments with the following results were made :— [table]
[Addition, May 27, after the reading of the paper.]—No. 1 has been kept at rest from May 17, while No. 2 has been kept oscillating more or less every day till yesterday, May 26, when both were oscillated, with the following results :— [table]
35. The investigation was continued with much smaller degrees of maximum angular distortion, to discover, if possible, the law of the molecular friction, the existence of which was demonstrated by these experiments. Two questions immediately occurred :—What is the law of subsidence of range in any single series of oscillations, the vibrator being undisturbed by external force? and (question (a) of § 33 above) what is the relation between the law of subsidence in two sets of oscillations having different periods, with the same elastic body in the same circumstances of elastic force, as for instance the same or similar metallic wires with equal weights hung upon them, performing torsional oscillations in different times on account of the moments of inertia of the suspended masses being different?
36. So far as the irregularities depending on previous conditions of the elastic substance allowed any simple law to be indicated, the experimental answer to the first question for degrees of angular distortion much smaller than the palpable limits of elasticity was the Compound Interest Law, that is to say,— The diminutions of range per equal intervals of time or per equal numbers of oscillations bore a constant proportion to the diminishing range; or, The differences of the logarithms of the ranges were proportional to the intervals of time.
The only approach to an answer to the second question yet obtained is that the proportionate losses of amplitude in the different cases are not such as they would be if the molecular resistance were simply proportional to the velocity of change of shape in the different cases. If the molecular friction followed this simple law, the proportionate diminutions of range per period would be directly as the square roots of the periods, or per equal intervals of time they would be inversely as the square roots of the periods. Instead of the proportion being so, the loss was greater with the longer periods than that calculated according to the law of square roots from its amount in the shorter periods. It was in fact as it would be if the result were wholly or partially due to imperfect elasticity, or “elastische Nach-wirkung” — elastic after-working — as the Germans call it (compare section 6 above). To form a rough idea of the results, irrespectively of the ultimate molecular theory (which is to be looked for in the proper extension of Maxwell’s kinetic theory of viscosity of gases), consider a perfectly elastic vesicular solid, whether like a sponge with communications between the vesicles, or with each vesicle separately inclosed in elastic solid: imagine its pores and interstices filled up with a viscous fluid, such as oil. Static experiments on such a solid will show perfect elasticity of bulk and shape; kinetic experiments will show losses of energy such as are really shown by vibrators of india-rubber, jelly, glass, metals, or other elastic homogeneous solids, but more regular, and following more closely the compound interest law for single series and the law of relation to square roots of periods stated above for sets of oscillations in different periods. In short, according to Stokes’s law of viscosity of fluids, our supposed vesicular vibrator would follow the law of subsidence of a simple vibrator experiencing a resistance simply proportional to the velocity of its motion, while no such simple law is applicable to the effects of the internal molecular resistance in a vibrating elastic solid.
37. Hooke's Law.— A law expressed by Hooke with Latin terseness in the words Ut tensio sic vis is the foundation of the mathematical theory of the elasticity of hard solids. By tensio here is meant not force (as is generally meant by the English word tension), but an elongation produced by force. In English, then, Hooke’s law is that elongation (understood of an elastic solid) is proportional to the force producing it. It is, of course, to be extended continuously from elongation to contraction in respect to the effect, and from pull to push in respect to the cause; and the experiments on which it is founded prove a perfect continuity from a pulling force to a smaller force in the same direction, and from the less force to zero, and from zero of pulling force to different degrees of push or positive pressure, or negative pull. Experimental proof merely of the continuity of the phenomena through zero of force suffices to show that, for infinitely small positive or negative pulls, positive or negative elongation is simply proportional to the positive or negative pull; or, in other words, positive or negative contraction is proportional to the positive or negative pressure producing it. But now must be invoked minutely accurate experimental measurement to find how nearly the law of simple proportionality holds through finite ranges of contraction and elongation. The answer happily for mathematicians and engineers is that Hooke's law is fulfilled, as accurately as any experiments hitherto made can tell, for all metals and hard solids each through the whole range within its limits of elasticity; and for woods, cork, india-rubber, jellies, when the elongation is not more than two or three per cent., or the angular distortion not more than a few hundredths of the radian (or not more than about two or three degrees). The same law holds for the condensation of liquids up to the highest pressures under which their compressibility has hitherto been accurately measured. [A decided but small deviation from Hooke’s law has been found in steel pianoforte wire under combined influence of torsion and longitudinal pull by Mr M t Farlane in experiments made for the present article after this section was in type. See section 81.]
Boyle’s law of the “spring of air v shows that the augmentation of density of a gas is simply proportional to the augmentation of the pressure, through the very wide ranges of pressure through which that law is approximately enough fulfilled. Hence the infinitesimal diminution of volume produced by a given infinitesimal augmentation of pressure varies as the square of the volume, and the proportionate diminution of volume (that is to say, the ratio of the diminution of volume to the volume) is proportional to the volume, or inversely proportional to the density. Andrews’s experiments on the compressibility of a fluid, such as carbonic acid, at temperatures slightly above the critical temperature, and of the gas and of liquids at temperatures slightly below the critical temperature, are intensely interesting, not merely in respect to the natural history of elasticity, but as opening vistas into the philosophy of molecular action.
We cannot expect to find any law of simple proportionality between stress and change of dimensions, or proportionate change of dimensions, in the case of any elastic or semi-elastic “soft” solids, such as cork on one hand or india-rubber or jellies on the other, when strained to large angular distortions, or to large proportionate changes of dimensions. The exceedingly imperfect elasticity of all these solids, and the want of definiteness of the substance of many of them, renders accurate experimenting unavailable for obtaining any very definite or consistent numerical results; but it is interesting to observe roughly the forces required to produce some of the great strains of which they are capable without any total break down of elastic quality; for instance, to hang weights successively on an indiarubber band and measure the elongations. This any one may readily do, and may be surprised to find the enormous increase of resistance to elongation presented by the attenuated band before it breaks.
38. Homogeneousness defined.— A body is called homogeneous when any two equal, similar parts of it, with corresponding lines parallel and turned towards the same parts, are un distinguishable from one another by any difference in quality. The perfect fulfilment of this condition, without any limit as to the smallness of the parts, though conceivable, is not generally regarded as probable, for any of the real solids or fluids known to us, however seemingly homogeneous. It is held by all naturalists that there is a molecular structure, according to which, in compound bodies such as water, ice, rock-crystal, &c., the constituent substances lie side, by side, or arranged in groups of finite dimensions, and even in bodies called simple (that is those not known to be chemically resolvable into other substances) there is no ultimate homogeneousness. In other words, the prevailing belief is that every kind of matter with which we are acquainted has a more or less coarse-grained texture, whether (as great masses of solid brick-work or stone-building, or as natural sandstone or granite rocks) having visible molecules, or (as seemingly homogeneous metals, or continuous crystals, or liquids, or gases) having molecules too small to be directly visible, or measurable but not undiscoverably small,—really, it is to be believed, of dimensions to be accurately determined in future advances of science. Practically the definition of homogeneousness may be applied on a very large scale to masses of building or coarse-grained conglomerate rock, or on a more moderate scale to blocks of common sandstone, or on a very small scale to seemingly homogeneous metals; or on a scale of extreme, undiscovered fineness, to vitreous bodies, continuous crystals, solidified gums, as india-rubber, gum-arabic, <fcc., and fluids.
39. Isotropic and Aeolotropic Substances defined.— The substance of a homogeneous solid is called isotropic when a spherical portion of it, tested by any physical agency, exhibits no difference iu quality however it is turned. Or, which amounts to the same, a cubical portion, cut from any position in an isotropic body, exhibits the same qualities relatively to each pair of parallel faces. Or two equal and similar portions cut from any positions in the body, not subject to the condition of parallelism (section 38),are undistinguishable from one another. A substance which is not isotropic, but exhibits differences of quality in different directions, is called aeolotropic: The remarks of section 38 relative to homogeneousness in the aggregate, and the supposed ultimately heterogeneous texture of all substances, however seemingly homogeneous, indicate corresponding limitations and non-rigorous practical interper-tations of isotropy and aeolotropy.
40. Isotropy and Aeolotropy of different sets of properties.— The substance of a homogeneous solid may be isotropic in one quality or class of qualities, but aeolotropic in others. Or a transparent substance may transmit light at different velocities in different directions through it (that is, be doubly-refracting), and yet a cube of it may (and does in many natural crystals) show no sensible difference in its absorption of white light transmitted across it perpendicularly to any of its three pairs of faces. Or (as a crystal which exhibits dichroism) it may be sensibly aeolotropic relatively to the absorption of light, but not sensibly double-refracting, or it may be dichroic and doubly-refracting, and yet it may conduct heat equally in all directions. Still, as a rule, a homogeneous substance which is aeolotropic for one quality must be more than infinitesimally aeolotropic for every quality which has directional character admitting of a corresponding aeolotropy.
4L Moduluses of Elasticity.— A modulus of elasticity is the number obtained by dividing the number expressing a stress by the number expressing the strain which it produces. A modulus is called a principal modulus when the stress is such that it produces a strain of its own type.
(1.) An isotropic solid has two principal moduluses— a modulus of compression and a rigidity.
(2.) A crystal of the cubic class (fluor-spar, for instance) has three principal moduluses,— one modulus of compression and two rigidities.
(3.) An aeolotropic solid having (what no natural crystal has, but what a drawn wire has) perfect isotropy of physical qualities relative to all lines perpendicular to a certain axis of its substance has three principal moduluses,— two determinable from its different compressibilities along and perpendicular to the axis, or from one compressibility and the “Young’s modulus” (section 42) of an axial bar of the substance, or determinable from two compressibilities; and one rigidity determinable by measurement of the torsional rigidity of a round axial bar of the substance.
(4.) A crystal of Iceland spar has four principal moduluses,—three like those of case (3), and another rigidity depending on (want of complete circular symmetry, and) possession of triple symmetry of form, involving sextuple elastic symmetry, round the crystalline axis.
(5.) A crystal of the rectangular parallelepiped (or “tessaral”) class has six distinct principal moduluses which, when the directions of the principal axes are known, are determinable by six single observations,—three, of the three (generally unequal) compressibilities along the three axes; and three, of the three rigidities (no doubt generally unequal) relatively to the three simple distortions of the parallelepiped, in any one of which one pair of parallel rectangular faces of the parallelepiped become oblique parallelograms.
(6.) An aeolotropic solid generally has six principal moduluses, which, when a piece of the solid is presented without information, and without any sure indication from its appearance of any particular axis or axes of •symmetry of any kind, require just twenty-one independent observations for the determination of the fifteen quantities specifying their types, and the six numerical values of the moduluses themselves.
42. “Young's Modulus,” or Modulus of Simple Longitudinal Stress.— Thomas Young called the -modulus of elasticity of an elastic solid thé amount of the end-pull or end-thrust required to produce any infinitesimal elongation or contraction of a wire, or bar, or column of the substance multiplied by the ratio of its length to the elongation or contraction. In this definition the definite article is clearly misapplied. There are, as we have seen, two moduluses of elasticity for an isotropic solid,—one measuring elasticity of bulk, the other measuring elasticity of shape. An interesting and instructive illustration of the confusion of ideas so often rising in physical science from faulty logic is to be found in “An Account of an Experiment on the Elasticity of Ice: By Benjamin Bevan, Esq.,, in a letter to Dr Thomas Young, Foreign Sec. R. S.” and in Young’s “Note” upon it, both published in the Transactions of the Royal Society for 1826. Bevan gives an interesting account of a well-designed and well-executed experiment on the flexure of a bar, 3^97 inches thick, 10 inches broad, and 100 inches long, of ice on a pond near Leighton Buzzard (the bar remaining attached by one end to the rest of the ice, but being cut free by a saw along its sides and across its other end), by which he obtained a fairly accurate determination of “the modulus of ice ”; and says that he repeated the experiment in various ways on ice bars of various dimensions, some remaining attached by one end, others completely detached, and found results agreeing with the first as nearly “as the admeasurement of the thickness could be ascertained.” He then proceeds to compare “the modulus of ice” which he had thus found with “the modulus of water,” which he quotes from Young’s Lectures as deduced from Canton’s experiments on the compressibility of water. Young in his “Note” does not point out that the two moduluses were essentially different, and that the modulus of his definition, the modulus determinable from the flexure of a bar, is essentially zero for every fluid. We now call “Young’s modulus” the particular modulus of elasticity defined as above by Young, and so avoid all confusion.
43. Modulus of Rigidity.— The “modulus of rigidity” of an isotropic solid is the amount of tangential stress divided by the deformation it produces,—the former being measured in units of force per unit of the area to which it is applied in the manner indicated by the annexed diagram (fig. 3), and the latter by the variation of each of the four right angles reckoned in fraction of the radian. By drawing either diagonal of the square in the diagram we see that the distorting stress represented by it gives rise to a normal traction on every surface of the substance perpendicular to the square and parallel to one of its diagonals, and an equal normal pη'essure on every surface of the solid perpendicular to the square and parallel to the other diagonal; and that the amount of each of these normal forces per unit of area is equal to the amount per unit area of the tangential forces which the diagram indicates. The corresponding geometrical proposition, also easily proved, is as follows: A strain compounded of a simple extension in one set of parallels, and a simple contraction of equal amount in any other set perpendicular to those, is the same as a simple shear in either of the two sets of planes cutting the two sets of parallels at 45°, and the numerical measuring of this shear or simple distortion is equal to double the amount of the elongation or contraction, each reckoned per unit of .length.
Hence we have another definition of “modulus of rigidity” equivalent to the preceding :—The modulus of rigidity of an isotropic substance is the amount of normal traction or pressure per unit of area, divided by twice the amount of elongation in the direction of the traction or of contraction in the direction of the pressure, when a piece of the substance is subjected to a stress producing uniform distortion.
44. Conditions fulfilled in Elastic Isotropy. —To be elastically isotropic, a spherical or cubical portion of any solid, if subjected to uniform normal pressure (positive or negative) all round, must, in yielding, experience no deformation, and therefore must be equally compressed (or dilated) in all directions. But, further, a cube cut from any position in it, and acted on by tangential or distorting stress in planes parallel to two pairs of its sides, must experience simple deformation, or “shearing” parallel to either pair of these sides, unaccompanied by condensation or dilatation,
and the same in amount for all the three ways in which s stress may be thus applied to any one cube, and for different cubes taken from any different positions in the solid. Hence the elastic quality of a perfectly elastic, homogeneous, isotropic solid is fully defined by two elements,—its resistance to distortion and its resistance to compression. The first has been already considered (section 43). The second is measured by the amount of uniform pressure in all directions per unit area of its surface required to produce a stated very small compression. The numerical reckoning of the first is the compressing pressure divided by the diminution of the bulk of a portion of the substance which, when uncompressed, occupies the unit volume. It is sometimes called the “elasticity of bull·,” or sometimes the “modulus of bulk-elasticity f sometimes the resistance to compression. Its reciprocal, or the amount of compression on unit of volume divided by the compressing pressure, or, as we may conveniently say, the compression per unit of volume per unit of compressing pressure, is commonly called the compressibility.
45. Strain produced by a single Longitudinal Stress (subject of Young’s Modulus').— Any stress whatever may be made up of simple longitudinal stresses. Hence, to find the relation between any stress and the strain produced by it, we have only to find the strain produced by a single longitudinal stress, which, for an isotropic solid, we may do at once thus :—A simple longitudinal stress P is equivalent to a uniform dilating tension ⅛P in all directions, compounded with two distorting stresses, each equal to ⅜∕ 3 , and having a common axis in the line of the given longitudinal stress, and their other two axes any two lines at right angles to one another and tc it. The diagram (fig. 4), drawn in a plane through one of these latter lines and the former, sufficiently indicates the synthesis,—the only forces not shown being those perpendicular to its plane.
Hence if n denote the rigidity, and k the modulus of compression, or the modulus of bulk-elasticity (being the same as the reciprocal of the compressibility), the effect will be an equal dilatation in all directions, amounting, per unit of volume, to
V (>λ
compounded with two equal distortions, each amounting to i ,Γ (s >∙
and having (section 43, footnote) their axes in the directions just stated for the axes of the distorting stresses.
46. lhe dilatation and two shears thus determined may be conveniently reduced to simple longitudinal strains by following the indications of section 43, thus :—
The two shears together constitute an elongation amount- 4P .
ln g tθ s -^ in the direction of the given force P, and equal ip contraction amounting to ’ '- in all directions perpendicular to it. And the cubic dilatation 'γ implies a lineal dilatation, equal in all directions, amounting to ~.
On the whole, therefore, we have linear elongation-p(-^ + <^), in the direction of the) applied stress, and
/1 ι∖? (3) linear con traction-Pi in all directions per-1
pendicular to the applied stress. J
47. Hence “Young’s Modulus” = , and when the
3⅛ + n ends of a column, bar, or wire of isotropic material are acted on by equal and opposite forces, it experiences a lateral lineal contraction equal to 2 βk+ n ) ^ ιe longitudinal dilatation, each reckoned as usual per unit of lineal measure. One specimen of the fallacious mathematics referred to in chap. xvi. of the mathematical theory below is a celebrated conclusion of Navier’s and Poisson’s that the ratio of lateral contraction to elongation by pull without transverse force is -∣. This would require the rigidity to be ⅜ of the resistance to compression, for all solids; which was first shown to be false by Stokes from many obvious observations, proving enormous discrepancies from it in many well-known bodies, and rendering it most improbable that there is any approach to a constancy of ratio between rigidity and resistance to compression in any class of solids. Thus clear elastic jellies and india-rubber present familiar specimens of isotropic homogeneous solids which, while differing very much from one another in rigidity (“stiffness ”), are probably all of very nearly the same compressibility as water, which is about per atmosphere.
Their resistance to compression, measured by the reciprocal of this, is obviously many hundred times the absolute amount of the rigidity of the stiffest of those substances. A column of any of them, therefore, when pressed together or pulled out, within its limits of elasticity, by balancing forces applied to its ends (or an india-rubber band when pulled out), experiences no sensible change of volume, though very sensible change of length. Hence the proportionate extension or contraction of any transverse diameter must be sensibly equal to half the longitudinal contraction or extension; and such substances may be practically regarded as incompressible elastic solids in interpreting all the phenomena for which they are most remarkable. Stokes gave reasons for believing that metals also have in general greater resistance to compression, in proportion to their rigidities, than according to the fallacious theory, although for them the discrepancy is very much less than for the gelatinous bodies. This probable conclusion was soon experimentally demonstrated by Wertheim, who found the ratio of lateral to longitudinal change of lineal dimensions, in columns acted on solely by longitudinal force, to be about ⅜ for glass and brass; and by Kirchhoff, who, by a well-devised experimental method, found , 387 as the value of that ratio for brass, and , 294 for iron. For copper it is shown to lie between ’ 226 and ’ 441, by experiments quoted below, measuring the torsional and longitudinal rigidities of copper wires.
48. All these results indicate rigidity less in proportion to the compressibility than according to Navier’s and Poisson’s theory. And it has been supposed by many naturalists who have seen the necessity of abandoning that theory as inapplicable to ordinary solids that it may be regarded as the proper theory for an ideal perfect solid, and as indicating an amount of rigidity not quite reached in any real substance, but approached to in some of the most rigid of natural solids (as, for instance, iron). But it is scarcely possible to hold a piece of cork in the hand without perceiving the fallaciousness of this last attempt to maintain a theory which never had any good foundation. By careful measurements on columns of cork of various forms (among them, cylindrical pieces cut in the ordinary way for bottles), before and after compressing them longitudinally in a Bramah’s press, we have found that the change of lateral dimensions is insensible both with small longitudinal contractions and return dilatations, within the limits of elasticity, and with such enormous longitudinal contractions as to ⅜ or ∣ of the original length. It is thus proved decisively that cork is much more rigid, while metals, glass, and gelatinous bodies are all less rigid, in proportion to resistance to compression, than the supposed “perfect solid”; and the practical invalidity of the theory is experimentally demonstrated. By obvious mechanism of jointed bars a solid may be designed which shall swell laterally when pulled, and shrink laterally when compressed, in one direction, and which shall be homogeneous in the same sense (article 40) as crystals and liquids are called homogeneous.
49. Modulus of Simple Longitudinal Strain.— In sections 45, 46, we examined the effect of a simple longitudinal stress in producing elongation in its own direction, and contraction in lines perpendicular to it. With stresses substituted for strains, and strains for stresses, we may apply the same process to investigate the longitudinal and lateral tractions required to produce a simple longitudinal strain (that is, an elongation in one direction, with no change of dimensions perpendicular to it) in a rod or solid of any shape.
Thus a simple longitudinal strain e is equivalent to a cubic dilatation e without change of figure (or linear dilatation ∣e equal in all directions), and two distortions consisting each of dilatation ⅜e in the given direction and contraction ⅜e in each of two directions perpendicular to it and to one another. To produce the cubic dilatation e alone requires (section 44) a normal traction ke equal in all directions. And, to produce either of the distortions simply, since the measure (section 43) of each is ⅛e, requires a distorting stress equal to n × ⅜e, which consists of tangential tractions each equal to this amount, positive (or drawing outwards) in the line of the given elongation, and negative (or pressing inwards) in the perpendicular direction. Thus we have in all normal traction ∙=(k + jη)e, in the direction of the given ) strain, and f , ,> normal traction == (k - ∣n)β, in every direction perpen-( ' '' dicular to the given strain. )
Hence the modulus of simple longitudinal strain is k + ⅜n.
50. Weight-Modulus and Length of Modulus.— Instead of reckoning moduluses in units of force per unit of area, it is sometimes convenient to express them in terms of the weight of unit bulk of the solid. A modulus thus reckoned, or, as it is called by some writers, the length of the modulus, is of course found by dividing the weight-modulus by the weight of the unit bulk. It is useful in many applications of the theory of elasticity, as, for instance, in this result, which is proved in the elementary dynamics of waves in an elastic solid or fluid (chap, xvii. of the Mathematical Theory, below) :—the velocity of transmission of longitudinal vibrations (as of sound) along a bar of cord, or of waves of simple distortion, or of simple longitudinal extension and contraction in a homogeneous isotropic solid, or of sound waves in a fluid, is equal to the velocity acquired by a body in falling from a height equal to half the length of the proper modulus for the 3⅛l ∣ - n ) f° r the first case, the modulus of rigidity (n) for the second, the modulus of simple longitudinal strain (k + ⅛n) for the third, the modulus of compression k for the fourth. Remark that for air the static “length-modulus of compression” at constant temperature is the same as what is often technically called the “height of the homogeneous atmosphere.”
51. In reckoning moduluses there must be a definite understanding as to the unit in terms of which the force is measured, which may be either the kinetic unit or the gravitation unit for a specified locality, that is, the weight in that locality of the unit of mass. Experimenters hitherto have stated their results in terms in the gravitation unit, each for his own locality,—the accuracy hitherto attained being scarcely in any cases sufficient to require corrections for the different intensities of gravity in the different places of observation.
The most useful and generally convenient specification of the modulus of elasticity of a substance is in grammes-weight per square centimetre. This has only to be divided by the specific gravity of the substance to give the length of the modulus. British measures, however, being still unhappily sometimes used in practical and even in scientific statements, we too often meet with reckonings of the modulus in pounds per square inch or per square foot, in tons per square inch, or of length of the modulus in feet or in British statute miles.
The reckoning most commonly adopted in British treatises on mechanics and practical statements is pounds per square inch. The modulus thus stated must be divided by the weight of 12 cubic inchee of the solid, or by the product of its specific gravity into ’ 4335, to find the length of the modulus in feet.
To reduce from pounds per square inch to grammes per square centimetre, multiply by 70*31, or divide by •014223. French engineers generally state their results in kilogrammes per square millimetre, and so bring them to more convenient numbers, being 10 0 1 0 ύ 0 of the inconveniently large numbers expressing moduluses in grammes weight per square centimetre, but it is much better to reckon in millions of grammes per square centimetre.
52. “Resilience” is a very useful word, introduced about forty years ago (when the doctrine of energy was beginning to become practically appreciated) by Lewis Gordon, first professor of engineering in the university of Glasgow, to denote the quantity of work that a spring (or elastic body) gives back when strained to,some stated limit and then allowed to return to the condition in -which it rests when free from stress. The word “resilience” used without special qualification may be understood as meaning extreme resilience, or the work given back by the spring after being strained to the extreme limit within which it can be strained again and again without breaking or taking a permanent set. In all cases for which Hooke’s law of simple proportionality between stress and strain holds, the resilience is obviously equal to the work done by a constant force of half the amount of the extreme force acting through a space equal to the extreme deflection.
53. When force is reckoned in “gravitation measure,” resilience per unit of the spring’s mass is simply the height that the spring itself, or an equal weight, could be lifted against gravity by an amount of work equal to that given back by the spring returning from the stressed condition.
54. Let the elastic body be a long homogeneous cylinder or prism with flat ends (a bar as we may call it for brevity), and let the stress for which its resilience is reckoned be positive normal pressures on its ends. The resilience per unit mass is equal to the greatest height from which the bar can fall with its length vertical, and impinge against a perfectly hard horizontal plane without suffering stress beyond its limits of elasticity. For in this case (as in the case of the direct impact of two equal and similar bars meeting with equal and opposite velocities, discussed in Thomson and Tait’s Natural Philosophy, section 303), the kinetic energy of the translational motion preceding the impact is, during the first half of the collision, wholly converted into potential energy of elastic force, which during the second half of the collision is wholly reconverted into kinetic energy of translational motion in the reverse direction. During the whole time of the collision the stopped end of the bar experiences a constant pressure, and at the middle of the collision the whole substance of the bar is for an instant at rest in the same state of compression as it would have permanently if in equilibrium under the influence of that pressure and an equal and opposite pressure on the other end. From the beginning to the middle of the collision the compression advances at a uniform rate through the bar from the stopped end to the free end. Every particle of the bar which the compression has not reached continues moving uniformly with the velocity of the whole before the collision until the compression reaches it, when it instantaneously comes to rest. The part of the bar which at any instant is all that is compressed remains at rest till the corresponding instant in the second half of the collision.
55. From our preceding view of a bar impinging against an ideal perfectly rigid plane, we see at once all that takes place in the real case of any rigorously direct longitudinal collision between two equal and similar elastic bars with flat ends. In this case the whole of the kinetic energy which the bodies had before collision reappears as purely translational kinetic energy after collision. The same would be approximately true of any two bars, provided the times taken by a pulse of simple longitudinal stress to run through their lengths are equal. Thus if the two bars be of the same substance, or of different substances having the same value for Young’s modulus, the lengths must be equal, but the diameters may be unequal. Or if the Young’s modulus be different in the two bars, their lengths must (Math. Theory, chap, xvii.) be inversely as the square roots of its values. To all such cases the laws of “collision between two perfectly elastic bodies,” whether of equal or unequal masses, as given in elementary dynamical treatises, are applicable. But in every other case part of the translational energy which the bodies have before collision is left in the shape of vibrations after collision, and the translational energy after collision is accordingly less than before collision. The losses of energy observed in common elementary dynamical experiments on collision between solid globes of the same substance are partly due to this cause. If they were wholly due to it they would be independent of the substance, when two globes of the same substance are used. They would bear the same proportion to the whole energy in every case of collision between two equal globes, or again, in every case of collision between two globes of any stated proportion of diameters, provided in each case the two which collide are of the same substances; but the proportion of translational energy converted into vibrations would not be the same for two equal globes as for two unequal globes. Hence when differences of proportionate losses of energy are found in experiments on different substances, as in Newton’s on globes of glass, iron, or compressed wool, this must be due to imperfect elasticity of the material. It is to be expected that careful experiments upon hard well-polished globes striking one another with such gentle forces as not to produce even at the point of contact any stress approaching to the limit of elasticity, will be found to give results in which the observed loss of translational energy can be almost wholly accounted for by vibrations remaining in the globes after collision.
56. Examples of Resilience.—Example 1.—In respect to simple longitudinal pull, the extreme resilience of steel pianoforte wire of the gauge and quality referred to in section 22 above (calculated by multiplying the breaking weight into half the elongation produced by it according to the experimental data of section 22) is 6066 metre-grammes (gravitation measure) per ten metres of the wire. Or, whatever the length of the wire, its resilience is equal to the work required to lift its weight through 179 metres.
Example 2.—The torsional resilience of the same wire, twisted in either direction as far as it can be without giving it any notable permanent set, was found to be equal to the work required to lift its weight through 1 '3 metres.
Example 3.—The extreme resilience of a vulcanized india-rubber band weighing 12 ’ 3 grammes was found to be equal to the work required to lift its weight through 1200 metres. This was found by stretching it by gradations of weights up to the breaking weight, representing the results by aid of a curve, and measuring its area to find the integral work given back by the spring after being stretched by a weight just short of the breaking weight.
57. Flexure of a Beam or Rod.— In the problem of simple flexure a bar or uniform rod or wire, straight when free from stress, is kept in a circular form by equal opposing couples properly applied to its ends. The parts of the bar on the convex side of the circle must obviously be stretched longitudinally, and those on the concave side contracted longitudinally, by the flexure. It is not obvious, however, what are the conditions affecting the lateral shrinkings and swellings of ideal filaments into which we may imagine the bar divided lengthwise. Earlier writers had assumed without proof that each filament, bent as it is in its actual position in the bar, is elongated or contracted by the same amount as it would be if it were detached, and subjected to the same end pull or end compression with its sides quite free to shrink or expand, but they had taken no account of the lateral shrinking or swelling which the filament must really experience in the bent bar. The subject first received satisfactory mathematical investigation from St Venant. He proved that the old supposition is substantially correct, with the important practical exception of the flat spring referred to in section 59 below. His theory shows that, in fact, if we imagine the whole rod divided parallel to its length into infinitesimal filaments, each of these shrinks or swells laterally with sensibly the same freedom as if it were separated from the rest of the substance and subjected to end pull or end compression, lengthening or shortening it in a straight line to the same extent as it is really lengthened or shortened in the circular arc which it becomes in the bent rod. He illustrates the distortion of the cross section by which these changes of lateral dimensions are necessarily accompanied in the annexed diagram (fig. 5), in which either the whole normal section of a rectangular beam, or a rectangular area in the normal section of a beam of any figure, is represented in its strained and unstrained figures, with the central point O common to the two. The flexure is in planes perpendicular to YOY 1 , and is concave upwards (or towards
X),—G, the centre of curvature, being in the direction indicated, but too far to be included in the diagram. The straight sides AC, BD, and all straight lines parallel to them, of the unstrained rectangular area become concentric arcs of circles concave in the opposite direction, their centre of curvature H being (articles 47, 48) for rods of india-rubber or gelatinous substance,or of glass or metal, from 2 to 4 times as far from O on one side as G is on the other. Thus the originally plane sides AC, BD of a rectangu lar bar become anticlastic surfaces, of curvatures — and P
— σ .
—, in the two principal sections, if σ denote the ratio of lateral shrinking to longitudinal extension. A flat rectangular, or a square, rod of india-rubber [for which σ amounts (section 47) to very nearly ∣, and which is susceptible of very great amounts of strain without utter loss of corresponding elastic action] exhibits this phenomenon remarkably well.
58. Limits to the bending of Pods or Beams of hard solid substance.— For hard solids, such as metals, stones, glasses, woods, ivory, vulcanite, papier-maché, elongations and contractions to be within the limits of elasticity must generally (section 23) be less than τ ) rσ . Hence the breadth or thickness of the bar in the plane of curvature must generally be less than - r 1 σ of the diameter of curvature in order that the bending may not break it, or give it a permanent bend, or strain it beyond its “limits of elasticity.”
59. Exceptional case of Thin fiat Spring, too much bent to fulfil conditions of section bl.— St Venant’s theory shows that a farther condition must be fulfilled if the ideal filaments are to have the freedom to shrink or expand as explained in section 57. For unless the breadth AC of the bar (or diameter perpendicular to the plane of flexure) be very small in comparison with the mean proportional between the radius OH and the thickness AB the distances from YY 1 to the corners A', C',would fall short of the half thickness, OE, and the distances to B', D', would exceed it, by differences comparable with its own amount. This would give rise to sensibly less and greater shortenings and stretchings in the filaments towards the corners than those supposed in the ordinary calculation of flexural rigidity (article 61), and so vitiate the result. Unhappily, mathematicians have not hitherto succeeded in solving, possibly not even tried to solve, the beautiful problem thus presented by the flexure of a broad very thin band (such as a watch spring) into a circle of radius comparable with a third proportional to its thickness and its breadth.
60. But, provided the radius of curvature of the flexure is not only a large multiple of the greatest diameter, but also of a third proportional to the diameters in and perpendicular to the plane of flexure; then, however great may be the ratio of the greatest diameter to the least, the preceding solution is applicable; and it is remarkable that the necessary distortion of the normal section (illustrated in the diagram of article 57) does not sensibly impede the free lateral contractions and expansions in the filaments, even in the case of a broad thin lamina (whether of precisely rectangular· section, or of unequal thicknesses in different parts).
61. Flexural Rigidities of a Pod or Beam.— The couple required to give unit curvature in any plane to a rod or beam is called its flexural rigidity for curvature in that plane. When the beam is of circular cross section and of isotropic material, the flexural rigidity is clearly the same, whatever be the plane of flexure through the axis, and the plane of the bending couple coincides with the plane of flexure. It might be expected that in a round bar of aeolotropic material, such as a wooden rod with the annual woody layers sensibly plane and parallel to a plane through its axis, would show different flexural rigidities in different planes,—in the case of wood, for example, different according as the flexure is in a parallel or perpendicular to the annual layers. This is not so, however; on the contrary, it is easy to show, by an extension of St Venant’s theory, that in the case of the wooden rod the flexural rigidity is equal in all planes through the axis, and that the plane of flexure always agrees with the plane of the bending couple, and to prove generally that the flexure of a bar of aeolotropic substance, and composed it may be of longitudinal filaments of heterogeneous materials, is precisely the same as if it were isotropic, and that its flexural rigidities are calculated by the same rule from its Young’s modulus, provided that the aeolotropy is not such as (section 81) to give rise to alteration of the angle between the length and any diameter perpendicular to the length when weight is hung on the rod, or on any longitudinal filament cut from it. Excluding then all cases in which there is any such oblique aeolotropy, we have a very simple theory for the flexure of bars of any substance, whether isotropic or aeolotropic, and whether homogeneous or not homogeneous through the cross section.
62. Principal Flexural Rigidities and Principal Planes of Flexure of a Beam.— The flexural rigidity of a rod is generally not equal in different directions, and the plane of flexure does not generally coincide with the plane of the bending couple. Thus a flat ruler is much more easily bent in a plane perpendicular to its breadth than in the plane of its breadth; and if we apply opposing couples to its two ends in any plane through its axis not either perpendicular or parallel to its breadth, it is obvious that the plane in which the flexure takes place will be more inclined to the plane of the breadth than to the plane of the bending couple. Very elementary statical theory, founded on St Venant’s conclusions of section 57, shows that, whatever the shape and the distribution of matter in the cross section of the bar, there are two planes at right angles to one another such that if the bar be bent in either of these planes the bending couple will coincide with the plane of flexure. These planes are called principal planes of flexure, and the rigidities of the bar for flexure in these planes are called its principal flexural rigidities. When the principal flexural rigidities are known the flexure of the bar in any plano oblique to the principal planes is readily found by supposing it to be bent in one of the principal planes and simultaneously in the other, and calculating separately the couples required to produce these two component flexures. The positions of the principal planes of flexure, the relative flexural rigidities, and the law of elongation and contraction in different parts of the cross section, are found according to the following simple rules :—
(1.) Imagine an infinitely thin plane disc of the same shape and size as the cross section loaded with matter in simple proportion to the Young’s modulus in different parts of the cross section. Let the quantity of matter per unit area on any point of the disc be equal to the Young’s modulus on the corresponding point of the rod when the material is heterogeneous: on the other hand, when the material is homogeneous it is more convenient to call the quantity of matter unity per unit area of the disc. Considering different axes in the plane of the disc through its centre of inertia, find the two principal axes of greatest and least moments of inertia, and find the moments of inertia round them.
(2.) In whatever plane the bar be bent it will experience neither elongation nor contraction in the filament which passes through the centres of inertia of the cross sections found according to rule
(1) , nor ’ in the diameter of the cross section perpendicular to the plane of flexure.
(3. ) Thus all the particles which experience neither elongation nor contraction lie in a surface cutting the plane oi flexure perpendiculari y through the centres of inertia of the cross sections. All the material on the outside of this cylindrical surface is elongated, and all on the interior is contracted, in simple proportion to distance from it: the amount of the elongation or contraction being in fact equal to distance from this neutral surface divided by the radius of its curvature.
(4. ) Hence it is obvious that the portions of the solid on the two sides of any cross section must experience mutual normal force, pulling them towards one another in the stretched part, and pressing them from one another in the condensed part, and that the amount of this negative or positive normal pressure per unit of area must be equal to the Young’s modulus at the place, multiplied into the ratio of its distance from the neutral line of the cross section to the radius of curvature.
The sum of these positive and negative forces over the whole area of the cross section is zero in virtue of condition
(2) . Their couple resultant has its axis perpendicular to the plane of curvature when this line is either of the principal axes (3) of the cross section; and its moment is clearly equal to the moment of inertia of the material disc (1) divided by the radius of curvature. Hence the principal flexural rigidities are simply equal to the principal moments of inertia of this disc; and the principal flexural planée are the planes through its principal axes and the length of the bar; or taking the quantity of matter per unit area of the disc unity for the case of a homogeneous bar, we have the rule that the principal rigidities are equal to the product of the Young’s modulus into the principal moments of inertia of the cross sectional areas, and the principal planes of flexure are the longitudinal planes through the principal axes of this area.
63. Law of Torsion.— One of the most beautiful applications of the general equations of internal equilibrium of an elastic solid hitherto made is that of M. de St Venant to “the torsion of prisms.” In this work the mathematical methods invented by Fourier for the solution of problems regarding conduction of heat have been most ingeniously and happily applied by St Venant to the problem of torsion. To reproduce St Venant’s mathematical investigation here would make this article too long (it occupies 227 quarto pages of the Mémoires des Savants Étrangers) ; but s. statement of some of the chief results is given (sections 65-72), not only on account of their strong scientific interest, but also because they are of great practical value in engineering; and the reader is referred to Thomson and Tait’s Natural Philosophy, sections 700-710, for the proofs and for further details regarding results, but much that is valuable and interesting is only to be found in St Venant’s original memoir.
64. Torsion Problem stated and Torsional Rigidity defined.— To one end of a long, straight prismatic rod, wire, or solid or hollow cylinder of any form, a given couple is applied in a plane perpendicular to the length, while the other end is held fast: it is required to find the degree of twist produced, and the distribution of strain and strese throughout the prism. The amount of the twist per unit length divided by the moment of the couple is called the torsional rigidity of the rod or prism. This definition is founded simply on the extension of Hooke’s law to torsion discovered experimentally by Coulomb, according to which a rod or wire when twisted within limits of torsional elasticity exerts a reactive couple in simple proportion to the angle through which one end is turned relatively to the other. The internal conditions to be satisfied in the torsion problem are that the resultant action between the substance on the two sides of any normal section is a couple, in the normal plane, equal to the given couple. This problem has not hitherto been attacked for aeolotropic solids. Even such a case as that of the round wooden rod (section 61) with annual layers sensibly parallel to a plane through its length, will, when twisted, experience a distribution of strain complicated much by its aeolotropy. The following statements of results are confined to rods of isotropic material.
65. Torsion of Circular Cylinder.— For a solid or hollow circular cylinder, the solution (given first, we believe, by Coulomb) obviously is that each circular normal section remains unchanged in its own dimensions, figure, and internal arrangement (so that every straight line of its particles remains a straight line of unchanged length), but is turned round the axis of the cylinder through such an angle as to give a uniform rate of twist equal to the applied couple divided by the product of the moment of inertia of the circular area (whether annular or complete to the centre) into the modulus of rigidity of the substance.
For, if we suppose the distribution of strain thus specified to be actually produced, by whatever application of stress is necessary, we have, in every part of the substance, a simple shear parallel to the normal section, and perpendicular to the radius through it. The elastic reaction against this requires, to balance it (section 43), a simple distorting stress consisting of forces in the normal section, directed as the shear, and others in planes through the axis, and directed parallel to the axis. The amount of the shear is, for parts of the substance at distance r from the axis, equal obviously to τr, if τ be the rate of twist reckoned in radians per unit of length of the cylinder. Hence the amount of the tangential force in either set of planes is nr per unit of area, if n be the rigidity of the substance. Hence there is no force between parts of the substance lying on the two sides of any element of any circular cylinder coaxal with the bounding cylinder or cylinders; and consequently no force is required on the cylindrical boundary to maintain the supposed state of strain. And the mutual action between the parts of the substance on the two sides of any normal plane section consists of force in this plane, directed perpendicular to the radius through each point, and amounting to nτr per unit of area. The moment of this distribution of force round the axis of the cylinder is (if dσ denote an element of the area) nτffdaT i , or the product of nτ 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 shape 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 PE along the tangent, from each point of the surface, to the point in which this line is cut by a perpendicular to it from O 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 PN and PT along the normal and tangent respectively. The whole, shear PK being equal to τr its component PN is equal to τr sin ω or t.PE. The corresponding component of the required stress is ητ. PE, 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 seen, 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 Venant 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 out the solution fora 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. “Hydrokiηetic Analogue to Torsion Problem. Extracted from Thomson and Tait, sections 704, 705. 1 — We take advantage of the identity of mathematical conditions in St Venant’s 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 η 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 thesame 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 each 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 any normal section is equivalent.
69. St Venant’s treatise abounds in beautiful and instructive graphical illustrations of his results, from which the following are selected :—
(1.) Elliptic Cylinder.— The plain and dotted curvilineal arcs are (fig. 7) “contour lines” (coupes topographiques') 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 diagram.
(2.) Contour lines for St Venant's il étoile à quatre points arrondis.'' — This diagram (fig. 8) shows the contour lines, in all respects as in case (1), for the case of a prism having for section the figure indicated. The portions of curve outside the continuous closed curve are merely indications of mathematical extensions irrelevant to the physical problem.
(3.) Contour lines of normal section of triangular przsm, as xvarped by torsion, shown as in case (1 ) (fig. 9.
(4.) Contour lines of normal sections of square prisms as warped by torsion (fig. 10).
(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 2 + y 2 - a(x i - 6∙r 2 y 2 + y i ) = 1 - a, with the indicated values for a. It is remarkable that the values a = 0'5 and a = -⅜(s ∕2 - 1) give 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 sum of principal Flexural Rigidities than according to false extension (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 a torsional rigidity less than this. How great the deficiency may be expected to be in cases in which the figure of the section presents projecting angles, or considerable prominences (which may be imagined from the hydrokinetic analogy given in section 67), has been pointed out 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 flexural. With remarkable ingenuity and mathematical skill he has drawn beautiful illustrations of this important practical principle from his algebraic and transcendental solutions.
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 circular 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 simpler cases the places of greatest distortion are those points 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 elliυtic cylinder the 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 thenarrow 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 (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 Reentrant Angles, or any places of too sharp concave curvature.— A solid of any elastic substance, isotropic or aeolotropic, 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 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).
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 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 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; 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 experimentally verified by Joule (“Thermodynamic Properties of Solids,” 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 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.
(5.) An india-rubber band suddenly drawn out ç 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 Manchester, 2d series, vol. i. p. 288), quoted by Joule in his paper on “Thermodynamic Properties of Solids” (Transactions 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 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.
74. The thermodynamic theory gives one formula by which the change of temperature in every such case may be calculated when the other physical properties are known :— z 3 ~ te P . where θ 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, the change of temperature θ 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; ρ, the density; and J, Joule’s equivalent (taken as 42400 centimetres).
In using the formula for a fluid,/? 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 Î3 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 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 ∕> 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. Mathematically 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 ^⅛mospheres 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 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
· · (1.)
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, θ the elevation of temperature.
F or the case of p a small fraction of P the formula gives 0-(i-l)∣i .... (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. (i = 289°) and the ordinary atmospheric pressure, and if it be compressed gradually up to 32 atmospheres, or dilated to ⅛ 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]
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 eØ, according to the notation of section 74, is the diminution of the strain duo to the change of temperature θ. 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 - cθ, or, with θ replaced by its value according to the formula of section 7 4, L- e te -P- M JKp ’
Dividing p by this expression we find for the kinetic modulus
M'= -
1 ⅛ 2 M JKp ’ Hence
M'_ 1
M ⅛ 2 M
JK p ’
76. For any substance, fluid or solid, it is easily proved, without thermodynamic theory, that
M' K
Μ “Ν’
where K denotes the thermal capacity of a stated quantity of the substance under constant stress, and N its thermal capacity under constant strain (or thermal capacity when the body is prevented from change of shape or change of volume). For permanent gases, and generally for fluids approximately fulfilling Boyle’s and Charles’s laws as said above, k is proved by thermodynamic theory to be approximately constant. Its value for all gases for which it has been measured differs largely from unity, and probably also for liquids generally (except water near its temperature of maximum density).
On the other hand, for solids whether the stress considered be uniform compression in all directions or of any M' jç
other type, the value of or differs but very little from unity; and both for solids and liquids it is far from constant at different temperatures (in the case of water it is zero at 3° ∙9 Cent., and varies as the square of the difference of the temperature from 3° - 9 at all events for moderate differences from this critical temperature, whether above or M' £ below it). The following tables show the value of γ^ ° r ÿ , and the value of θ by the formula of sec. 74, for different fluid aud solid substances at the temperature 15° Cent. (289° absolute scale). The first table is for compression uniform in all directions; the second, necessarily confined to solids, is for the stress dealt with in “Young’s Modulus,” that is, normal pressure (positive or negative) on one set of parallel planes, with perfect freedom to expand or contract in all directions in these planes. A wire or rod pulled longitudinally is a practical application of the latter. [table] [table]
77. Experimental Results.— The following tables show determinations of moduluses of compression, of Young’s modulus, and of moduluses of rigidity by various experimenters and various methods. It will be seen that the Young’s moduluses obtained by Wertheim by vibrations, longitudinal or transverse, are generally in excess of those which he found by static extension; but the differences are enormously greater than those due to the heating and cooling effects of elongation and contraction (section 76), and are to be certainly reckoned as errors of observation. It is probable that his moduluses determined by static elongation are minutely accurate; the discrepancies of those found by vibrations are probably due to imperfections of the arrangements for carrying out the vibrational method:— [table] [table]
[table]
78. A question of great importance in the physical theory of the elasticity of solids, “What changes are produced in the moduluses of elasticity by permanent changes in its molecular condition,” has occupied the attention, no doubt, of every “naturalist” who has studied the subject, and valuable contributions to its answer by experiment had been given by Wertheim and other investigators, but solely with reference to Young’s modulus. In 1865 an investigation of the effect on the torsional rigidity of wires of different metals, produced by stretching them longitudinally beyond their limits of elasticity, was commenced in the physical laboratory of the university of Glasgow in its old buildings in 1865. The following description of experiments and table of results is extracted from the paper by W. Thomson “On the Elasticity and Viscosity of Metals,” already quoted (section 30), with reference to viscosity and fatigue of elasticity.
“Todetermine rigidities by torsional vibrations, taking advantage of an obvious but most valuable suggestion made to me by Dr Joule, I used as vibrator in each case a thin cylinder of sheet brass, turned true outside and inside (of which the radius of gravitation must be, to a very close degree of approximation, the arithmetic mean of the radii of the outer and inner cylindrical surfaces), supported by a thin flat rectangular bar, of which the square√)f the radius of gravitation is one-third of the square of the distance from the centre to the corner. The wire to be tested passed perpendicularly through a hole in the middle of the bar, and was there lirmly soldered. The cylinder was tied to the middle of the bar by light silk thread so as to hang with its axis vertical. Each wire, after having been suspended and stretched with just force enough to make it as nearly straight as was necessary for accuracy, was vibrated. Then it was stretched by hand (applied to the cross bar soldered to its lower end) and vibrated again, and stretched again, and so on till it broke.” The experiments were performed with great care and accuracy by Mr Donald M'Farlane. “The results, as shown in the accompanying table, were most surprising.”
The highest and lowest rigidities found for copper in the table are as follows :—
Highest rigidity 473 × 10 δ , being that of a wire which had been softened by heating it to redness and plunging it into water, and which was found to be of density 8'91.
Lowest rigidity 393’4 × 10 δ , being that of a wire which had been rendered so brittle by heating it to redness surrounded by potvdered charcoal in a crucible and letting it cool very slowly, that it could scarcely be touched without breaking it, and which had been found to be reduced in density by this process to as low as 8'674. The wires used were all commercial specimens—those of copper being all, or nearly all, cut from hanks supplied by the Gutta Percha Company, having been selected as of high electric conductivity, and of good mechanical quality, for submarine cables.
It ought to be remarked that the change of molecular condition produced by permanently stretching a wire or solid cylinder of metal is certainly a change from a condition which, if originally isotropic, becomes aeolotropic as to some qualities, and that the changed conditions may therefore be presumed to be aeolotropic as to elasticity. If so, the rigidities corresponding to the direct and diagonal distortions (indicated by No. 1 and No. 2 in fig. 14) must in all probability become different from one another when a wire is permanently stretched, instead of being equal as they must be when its substance is isotropic. It becomes, therefore, a question of extreme interest to find whether rigidity No. 2 is not increased by this process, which, as is proved by the experiments above described, diminishes, to a veιy remarkable degree, the rigidity No. 1. The most obvious experiment, and indeed the only practicable experiment, adapted to answer this question, for a wire or round bar is that of Cagniard-Latour, in which an accurate determination of the difference produced in the volume of the substance is made by applying and removing longitudinal traction within its limits of elasticity. With the requisite apparatus, which must be much more accurate than that of Cagniard-Latour, a most important and interesting investigation might be made. The results, along with an accurate determination of the Young’s modulus for the particular case, give (sec. 47) the modulus of compression, and the rigidity No. 2. Régnault suggested the use of hollow instead of solid cylinders, to be subjected to longitudinal pull, and (after the manner of the bulb and tube of a thermometer) a capillary tube to aid in measuring changes of volume of the hollow; and Wertheim, adopting this excellent suggestion, obtained seemingly very accurate results for brass and glass, which are given in the tables of section 77. [table]
79. The following tables show the effects of differences of temperature on the Young’s Modulus, rigidity-modulus, and modulus of compressibility of various substances :— [table]
The above results are from Wertheim’s “Mémoires” on Elasticity, Aim. de Chim. et Phys., tom xii. (1844).
The change in the rigidity-modulus produced by change of temperature was investigated by Kohlrausch. He found that it is expressed by the formula n = η 0 (1 — at - βt 2 ), where ?? 0 denotes the value of the rigidity-modulus at 0° C., n its value at temperature t, and α, β coefficients the values of which for iron, copper, and brass are as follows:— [table] [table]
80. Tempering soft iron by long-continued stress.— Preliminary experiments by Mr J. T. Bottomley towards the investigation promised in section 5 above have discovered a very remarkable property of soft iron wire respecting its ultimate tensile strength. Eight different specimens, tested by the gradual application of more and more weight within ten minutes of time in each case until the wire broke, boro from 43⅛∙ to 46 lb (average 45 - 2) just before breaking, with elongations of from 17 per cent to 22 per cent. Another specimen left with 43 lb hanging on it for 24 hours, and then tested by the gradual addition of weights during 25 minutes till it broke, bore 49∣ lb before breaking, with elongation of 15 per cent. Another left for 3 days 11 hours 40 minutes with 43 lb hanging on it, and then tested by the gradual addition of weights during 31 minutes till it broke, bore 51⅜ lb just before breaking, with elongation of 14 ’ 4 per cent. Another specimen of the same wire was set up with 40 lb hanging on it on the 5th of July 1877, on the Gth of July 3 It) were added, on the 9th 1⅛ lb more, and on the 10th ∣ lb more, making in all on this date 451 lb. Thenceforward day by day, with occasional intervals of two days or three days, the weight was increased first by half a pound at a time, and latterly by a quarter of a pound at a time, until on the 3d of September the wire broke with 57∣ lb (elongation not recorded). This gradual addition of weight therefore had increased the tensile strength of the metal by 26'7 per cent.!
81. Experiments made for this article.· —There are many subjects in the theory of elasticity regarding which information to be obtained by experiment only is greatly wanted. Several of these have been pointed out above (section 21), and while this article was being put in type, experiments were made in the physical laboratory of the university of Glasgow with a view of answering some of the questions proposed. Mr Donald M i Farlane, besides making the experiments referred to insections 3 and 21, investigated the effects of applying different amounts of pull to a steel pianoforte wire which had been twisted to nearly its limits of elasticity, and which was kept twisted by means of a couple. The results proved a deviation from Hooke’s law by showing a diminution of the torsional rigidity, of about 1’6 per cent., produced by hanging a weight of 112 lb on the wire. Of this 1 , 2 per cent, is accounted for by elongation and by shrinkage of the diameter, leaving ’ 4 per cent, of diminution of the rigidity-modulus.
It was also found that when the wire was twisted far beyond its limits of elasticity, and then freed from torsional stress, a weight hung on it caused it to untwist slightly. When the weight was removed and reapplied again and again, the lower end of the wire always turned in the same direction as the permanent twist when the weight was removed, and in the opposite direction when it was applied. This result shows the development of aeolotropic quality in the substance of the wire, according to which a small cube cut from any part of it far out from the axis, with two sides of the cube parallel to the length, and the other two pairs of sides making angles of 45° with the length, would show different compressibilities in the directions perpendicular to the last-mentioned pairs of sides.
Another very interesting result, discovered in the course of these experiments, was that when a length of five metres of the steel wire, with a weight of 39 lb hung upon it, was twisted to the extent of 95 turns, it became gradually elongated to the extent of - 1 ⅞ 1 0 σ the length of the wire; when farther twisted it began to shorten till, when 25 turns had been given (in all 120 turns), the weight had risen from its lowest position through nearly 7yo ι 0 - σ of the length of the wire, so that the previous elongation had been diminished by about ∣ of its amount.
Experiments were also made by Mr Andrew Gray and Mr Thomas Gray for the purpose of determining the effects of various amounts of permanent twist in altering the rigidity-modulus and the Young’s modulus of wires of copper, iron, and steel. A copper wire, of 3 ’ 15 metres in length and ’ 154 centimetre diameter, No. 17 B.W.G., which had a rigidity-modulus of 442 million grammes per square centimetre to begin with, was found to have 420 after 10 turns, showing a diminution in the modulus of √ σ of its own amount. The diminution went on rapidly until 100 turns of permanent twist had been given, when the modulus was·as low as 385. The diminution of the modulus continued with further twist, but very slowly, up to 1225 turns, when the modulus was found to be 371, showing a diminution to the extent of ⅜ of its original value! There was little farther change until 1400 turns had been given, when the modulus began to increase. At 1525 turns its value was 373, and at 1625 it was 377. Twenty turns more broke the wire before the torsional elasticity had been again determined.
A piece of iron wire of nearly the same length, about three metres, but of smaller diameter (’ 087 centimetre), showed continued diminution of torsional rigidity as far as 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 torsional rigidity had been made.
The steel pianoforte wire also showed a diminution of torsional 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 lb, 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, while 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 lb, was in one of Mr M i Farlane's experiments twisted 120 turns, and then allowed to untwist, and 38∣ turns came out, leaving the wire in equilibrium with 81'∣ 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 THEORY OF ELASTICITY. Part I.— On Stresses and Strains. 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.
Def. 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 experience 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 “to strain.” A ship is said to “strain” if in launching, or when working in a heavy sea, the different parts of it experience relative motions.
Cπapter II.— Homogeneous Stresses and Homogeneous Strains. Def. A stress is said to be homogeneous throughout a body when equal and similar portions of the body, with corresponding lines parallel, experience equal and parallel pressures or tensions on corresponding elements of their surfaces.
Cor. When a body is subjected to any homogeneous stress, thθ 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 pιanes remain in parallel 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 circumstances preventing the ends from expanding laterally to the same extent as tlιe 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 heterogeneous as to intensity, from the axis to the surface of a cylindrical wire under torsion, and 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 sides.
For let P(X), P(Y), P(Z) denote the components, parallel to X, Y, Z, any three rectangular lines of reference, of the force experienced per unit of surface at any portion of the solid bounded by a plane parallel to (Y, Z); Q(X), Q(Y), Q(Z), the corresponding components of the force experienced by any surface of the solid parallel to (Z, X); and E(X), Ii(Y), B(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 R(Y) on the faces perpendicular to Z. Hence we must have
Q(Z) = R(Y).
Similarly it is seen that
, R(X) = P(Z)
and
P(Y) =Q(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 O parallel to the planes of the pairs of lines of reference, and a third plane K cutting these at angles a, ft, y respectively; so that as regards the areas of the different sides we shall have
A = K cos a, B = K cos ft, C = K cos γ.
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 resultant, 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 l, m, n, we have
F.K.J -P(X)A + T(Z)B + T(Y)C, F.K. m=T(Z)A+Q(Y)B+T(X)C, F.K.n =T(Y)A+T(X)B + R(Z)Ci and, by the relations between the cases stated above, we deduce
FI =P(X) cosα⅛T(Z) cos∕3 + T(Y) cosy Frn=T(Z) cos a + Q(Y) cos∕3÷T(X) cosy Fn =T(Y) cos <i⅛T(X) cos /3 4- R(Z) cos y
Hence the problem of finding (a, ft, y), so that the force F (l, m, n) may be perpendicular to it, will be solved by substituting cos a, cos ft, cos y for l, m, n in these equations. By the elimination of cos a, cos ft, cos y from the three equations thus obtained, we have the well-known cubic determinantal 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.
Bef. The three lines thus proved to exist for every possible homogeneous stress are called its axes. The planes of their pairs are called its normal planes; the mutual forces between parts of the body separated by these planes, or the forces on portions of the bounding surface parallel to them, are called the principal tensions.
Cor. 1. The Principal Tensions of the stress are the roots of the determinant cubic referred to in the demonstration.
Cor. 2. If a stress be specified by the notation P(X), &c., as explained above, its normal planes are the principal υlanes of the surface of the second degree whose equation is
P(X)X 2 +Q(Y)Y 2 +B(Z)Z 2 +2T(X)YZ+2T(Y)ZX
and its principal tensions are equal to the reciprocals of the squares of the lengths of the semi-principal-axes of the same surface (quantities which are negative of course for the principal axis or axes which do not cut the surface when the surface is a hyperboloid of one or of two sheets).
Cor. 3. The ellipsoid whose equation, referred to the rectangular axes if a stress, is
(1 - 2cF)X 2 +(l - 2cG)Y 2 +(l - 2eH)Z 2 = 1, where F, G, H denote the principal tensions, and e any infinitely small quantity, represents the stress, in the following manner :—
From any point P in the surface of the ellipsoid draw a line in the tangent plane half-way to the point where this plane is cut by a perpendicular to it through the centre; and from the end of the first-mentioned line draw a radial line to meet the surface of a sphere of unit radius concentric with the ellipsoid. The tension at this point of the surface of a sphere of the solid is in the line from it to the point P; and its amount per unit of surface is equal to the length of that infinitely small line, divided by e. Cor. 4. Any stress is fully specified by six quantities, viz., its three principal tensions (F, G, H), and three angles (0, φ, ψ) or three numerical quantities equivalent to the nine direction cosines specifying its axes.
Chapter IV.— On the Distribution of Displacement in a Strain. Prop. In eveιy homogeneous strain any part of the solid bounded by an ellipsoid remains bounded by an ellipsoid.
For all particles of the solid in a plane remain in a plane, and two parallel planes remain parallel. Consequently every system of conjugate diametral planes of an ellipsoid of the solid retain the property of conjugate diametral planes with reference to the altered curve surface containing the same particles. This altered surface is therefore an ellipsoid.
Prop. There is a single system (and only a single system, except in the cases of symmetry) of three rectangular planes for every homogeneous strain, which remain at right angles to one another in the altered solid.
Def. 1. These three planes are called the normal planes of the strain, or simply the strain-normals. Their lines of intersection are called the axes of the strain. The elongations of the solid per unit of length along these axes or perpendicular to these planes are called the Principal Elongations of the strain.
Piemark. The preceding propositions and definitions are not limited to infinitely small strains, but are applicable to whatever extent the body may be strained.
Prop. If a body, while experiencing an infinitely small strain, be held with one point fixed and the normal planes of the strain parallel to three fixed rectangular planes through the point 0, a sphere of the solid of unit radius having this point for its centre becomes, when strained, an ellipsoid, whose equation, referred to the strain-normals through 0, is
(1—2∙r)X 2 + (l-2y)Y 2 + (1—2z)Z 2 = 1, if X, y, z denote the elongations of the solid per unit of length, in the directions respectively perpendicular to these three planes; and the position, on the surface of this ellipsoid, attained by any particular point of the solid, is such that if a line be drawn in the tangent plane, half-way to the point of intersection of this plane with a perpendicular from the centre, a radial line drawn through its extremity cuts the primitive spherical surface in the primitive position of that point.
Cor. 1. For every stress, there is a certain infinitely small strain, and conversely, for every infinitely small strain, there is a certain stress, so related that if, while the strain is being acquired, the centre and the strain-normals through it are unmoved, the absolute displacements of particles belonging to a spherical surface of the solid represent, in intensity (according to a definite convention as to units for the representation* of force by lines) and in direction, the force (reckoned as to intensity, in amount per unit of area) experienced by the enclosed sphere of the solid, at the different parts of its surface, when subjected to the stress.
Cor. 2. Any strain is fully specified by six quantities, viz., its three principal elongations, and three angles (0, φ, ψ), or nine direction cosines, equivalent to three independent quantities specifying its axes.. . i Def. 2. A stress and an infinitely small strain related in the manner defined in Cor. 1, are said to be of the same type. The ellipsoid by means of which the distribution of force over the surface of a sphere of unit radius is represented in one case, and by means of which the displacements of particles from the spherical surface are shown in the other, may be called the geometrical type of either.
Cor. Any stress-or strain-type is fully specified by five quantities, viz., two ratios between its principal strains or elongations and three quantities specifying the angular position of its axes.
Chapter V.— Conditions of Perfect Concurrence between Stresses and Strains. Def. 1. Two stresses are said to be coincident in direction, or to be perfectly concurrent, when they only differ in absolute magnitude. The same relative designations are applied to two strains differing from one another only in absolute magnitude.
Cor. If two stresses or two strains differ by one being reverse to the other, they may be said to be negatively coincident in direction, or to be directly opposed or directly contrary to one another.
Def. 2. When a homogeneous stress is such that the normal component of the mutual force between the parts of the body on the two sides of any plane whatever through it is proportional to the augmentation of distance between the same plane and another parallel to it and initially at unity of distance, due to a certain strain experienced by the same body, the stress and the strain are said to be perfectly concurrent; also to be coincident in direction. The body is said to be yielding directly to a stress applied to it, when it is acquiring a strain thus related to the stress; and in the same circumstances, the stress is said to be working directly on the body, or to be acting in the same direction as the strain.
Cor. 1. Perfectly concurrent stresses and strains arc of the same type. ' - Cor. 2. If a strain is of the same type as the stress, its reverse will be said to be negatively of the same type, or to be directly opposed to the strain. A body is said to be working directly against a stress applied to it when it is acquiring a strain directly opposed to the stress; and in the same circumstances, the matter round the body is said to be yielding directly 7 to the reactive stress of the body upon it.
Chapter VI.— Orthogonal Stresses and Strains. Def. 1. A stress is said to act right across a strain, or to act orthogonally to a strain, or to be orthogonal to a strain, if work is neither done upon nor by the body in virtue of the action of the stress upon it while it is acquiring the strain.
Def. 2. Two stresses are said to be orthogonal when either coincides in direction with a strain orthogonal to the other.
Def. 3. Two strains are said to be orthogonal when either coincides in direction with a stress orthogonal to the other.
Examples— (1) A uniform cubical compression, and any strain involving no alteration of volume, are orthogonal to one another.
(2) A simple extension or contraction in parallel lines unaccompanied by any transverse .extension or contraction, that is, “.'»'simple longitudinal strain,” is orthogonal to any similar strain in lines at-right angles to those parallels.
(3) A simple longitudinal strain is orthogonal to a “simple tangential strain” in which the sliding is parallel to its direction or at right angles to it.
(4) Two infinitely small simple tangential strains in the same plane, with their directions of sliding mutually inclined at an angle of 45”, are orthogonal to one another.
(5) An infinitely small simple tangential strain is orthogonal to every infinitely small simple tangential strain, in a plane either parallel to its plane of sliding or perpendicular to its line of sliding.
Chapter VII.— Composition and Resolution of Stresses and of Strains.
Any number of simultaneously applied homogeneous stresses are equivalent to a single homogeneous stress which is called their resultant. Any number of superimposed homogeneous strains are equivalent to a single homogeneous resultant strain. Infinitely small strains may be independently 7 superimposed; and in what follows it will be uniformly understood that the strains spoken of are infinitely small, unless the contrary is stated.
Examples.— (1) A strain consisting simply of elongation in one set of parallel lines, and a strain consisting of equal contraction in a direction at right angles.to it, applied together, constitute a single strain, of the kind which that described in Example (3) of the preceding chapter is when infinitely small, and is called a plane distortion, or a simple distortion. It is also sometimes called a simple tangential strain, and when so considered, its plane of sliding may be regarded as either of the planes bisecting the angles between planes normal to the lines of the component longitudinal strains. . . .
(2) Any two simple distortions in one plane may be reduced to a single simpla distortion in the same plane. , . ,. .
(3) Two simple distortions not in the same plane have for their resultant a strain which is a distortion unaccompanied by change of volume, and which may be called a compound distortion. ...... .
(4) Three equal longitudinal elongations or condensations in three direction& et right angles to one another are equivalent to a single dilatation or condensation equal in all directions. The single stress equivalent to three equal tensions or pressures in directions at right angles to one another is a negative or positive pressure equal in all directions.
(5) If a certain stress or infinitely small strain be defined (Chapter III. Cor. 3, or Chapter IV.) by the ellipsoid
(1÷A) X 2 + (1+B)Y 2 +(1+C)Z 2 +DYZ+EZX+FXY=1, and another stress or infinitely small strain by the ellipsoid; (1+A')X 2 +(1+B')Y≈+(1+C')Z2+DΥZ+E'ZX+F'XY=1.
where A, B, C, D, E, F, &c., are all infinitely small, their resultant stress or strain is that represented by the ellipsoid α+ A+A')X 2 +(l+B+B')Y 2 +(l+C+C')Z≈+(D+D')YZ+(E+E')ZX +(F÷F')XY=1.
Chapter VIII.— Specification of Strains and Stresses by their Components according to chosen Types. Prop. Six stresses or six strains of six distinct arbitrarily chosen types may be determined to fulfil the condition of having a given stress or a given strain for their resultant, provided those six types are so chosen that a strain belonging to any one of them cannot be the resultant of any strains whatever belonging to the others.
For, just six independent parameters being required to express any stress or strain whatever, the resultant of any set of stresses or strains may be made identical with a given stress or strain by fulfilling six equations among the parameters which they involve; and therefore the magnitudes of six stresses or strains belonging to the six arbitarily chosen types may be determined, if their resultant be assumed to be identical with the given stress or strain.
Cor. Any stress or strain may be numerically specified in terms of numbers expressing the amounts of six stresses or strains of six arbitrarily chosen types which have it for their resultant.
Types arbitrarily chosen for this purpose will be called types ot reference. The specifying elements of a stress or strain will be called its components according to types of reference. The specifying elements of a strain may also be called its coordinates, with reference to the chosen types.
Examples.— (1) six strains in each of which one of the six edges of a tetrahedron of the solid is elongated while the others remain unchanged, may be used as types of reference for the specification of any kind of strain or stress. The ellipsoid representing any one of those six types will have its two circular sections parallel to the faces of the tetrahedron which do not contain ."he stretched side.
(2) Six strains consisting, any one of them, of an infinitely small alteration either of one of the three edges, or of one of the three angles between the faces, of a parallelepiped of the solid, while the other five angles and edges remain unchanged, may be taken as types of reference, for the specification of either stresses or strains. In some cases, as for instance in expressing the probable elastic properties of a crystal of Iceland spar, it might possibly be convenient to use an oblique parallelepiped for such a system of types of reference; but more frequently it will be convenient to adopt a system of types related to the deformations of a cube of the solid.
Chapter IX.— Orthogonal Types oj Reference. Def. A normal system of types of reference is one in which the strains or stresses of the different types are all six mutually orthogonal (fifteen conditions). A normal system of types of reference may also be called an orthogonal system. The elements specifying, with reference to such a system, any stress or strain, will be called orthogonal components or orthogonal coordinates.
Examples.— (1) The six types described in Example (2) of Chapter VIII. are clearly orthogonal, if the parallelepiped referred to is rectangular. Three of these are simple longitudinal extensions, parallel to the three sets of rectangular edges of the parallelepiped. The remaining three are plane distortions parallel to the faces, their axes bisecting the angles between the edges. They constitute the system of types of reference uniformly used hitherto by writers on the theory of elasticity.
<2) The six strains in which a spherical portion of the solid is changed into ellipsoids having the following equations—
(H-A)X 2 +Y 2 +Z 2 =) X 2 +(1+B)Y 2 +Z 2 =1 X 2 +Y 2 +(1+C)Z 2 =1 X 2 +Y 2 +Z 2 ⅛DYZ = 1 X 2 +Y 2 -f-Z 2 -f-EZX = l X 2 +Y 2 +Z 2 +FXY = 1, are of the same kind as those considered in the preceding example, and therefore constitute a normal system of types of reference. The resultant of the strains specified, according to those equations, by the elements A, B, C, D, E, F, is a strain in which the sphere becomes an ellipsoid whose equation—see above, Chapter VII. Ex. (5)—is
(1+A)X 2 +(1+B)Y 2 +(1+C)Z 2 +DYZ+EZX+FXY = 1.
(3) A compression equal in all directions (I.), three simple distortions having their planes at right angles to one another and their axes bisecting the angles between the lines of intersection of these planes (II.) (III.) (IV.), any simple or compound distortion consisting of a combination of longitudinal strains parallel to those lines of intersections (V.), and the distortion (VI.), constituted from the same elements which is orthogonal to the last, afford a system of six mutually orthogonal types which will be used as types of reference below in expressing the elasticity of cubically isotropic solids. (Compare Chapter X. Example 7 below.)
Chapter X.— On the Measurement of Strains and Stresses Def. Strains of any types are said to be to one another in the same ratios as stresses of the same types respectively, when any particular plane of the solid acquires, relatively to another plftne parallel to it, motions in virtue of those strains which are to one another in the same ratios as the normal components of the forces between the parts of the solid on the two sides of either plane due to the respective stresses.
Def. The magnitude of a stress and of a strain of the same type are quantities which, multiplied one by the other, give the work done on unity of volume of a body acted on by the stress while acquiring the strain.
Cor. 1. If X, y, z, ξ, η, £ denote orthogonal components of a certain strain, and if P, Q, 1i, S, T, U denote components, of the same type respectively, of a stress applied to a body while acquiring that strain, the work done upon it per unit of its volume will be
Pz + Qy+ Ite-J-Sf-f-T>j + Uζ.
Cor. 2. The condition that two strains or stresses specified by (x, y> ~> ⅛ m 0 an d (aj, , l∕> s l, c> n't O> terms of a normal system of types of reference, may be orthogonal to one another is χχ! +yy'+∞ ’ + ξζ'+η η' +ζζ'=0.
Cor. 3. The magnitude of the resultant of two, three, four, five, or six mutually orthogonal strains or stresses is equal to the square root of the sum of their squares. For if P, Q, &c., denote several orthogonal stresses, and F the magnitude of their resultant; and X, y, &c., a set of proportional strains of the same types respectively, and r the magnitude of the single equivalent strain, the resultant stress and strain will be of one type, and therefore the work done by the resultant stress will be Fr. But the amounts done by the several components will be Px, Qy, &c., and therefore
Fr=Px4-Qy+<fcc.
Now we have, to express the proportionality of the stresses and strains,
? = 9 = &c . =?
X y r∙
Each member must be equal to
P 2 + Q 2 + <⅛c..
p ∙r + Qy + *c. ’
and also equal to
P·r + (⅛ + ⅛c.
αr 2 + y 2 + <fcc. ∙
P2 4. Λ2 A. ,(∙ e
Hence r t ∙~ ' > which gives F 2 = P 2 + Q 2 + &c.
and r~ z 2 + y 2 + <⅛ c ., which gives r 2 = x 2 + y* + &c.
Cor. 4. A definite stress of some particular type chosen arbitrarily may be called unity; and then the numerical reckoning of all strains and stresses becomes perfectly definite.
Def. A uniform pressure or tension in parallel lines, amounting in intensity to the unit of force per unit of area normal to it, will be called a stress of unit magnitude, and will be reckoned as positive when it is tension, and negative when pressure.
Examples.— (1) Hence the magnitude of a simple longitudinal strain, in which lines of the body parallel to a certain direction experience elongation to an extent bearing the ratio κ to their'original dimensions, must be called κ.
(2) The magnitude of the single stress equivalent to three simple pressures in directions at right angles to one another each unity is —√3j a uniform compression in all directions of unity per unit of surface is a negative stress equal to √3 in absolute value.
(3) A uniform dilatation in all directions, in which lineal dimensions are augmented in the ratio 1:1 + x, is a strain equal in magnitude to z√3; or a uni-
E · form “cubic expansion” E is a strain equal to çyjj e
(4) A stress compounded of unit pressure in one direction and an equal tension in a direction at right angles to it, or which is the same thing, a stress compounded of two balancing couples of unit tangential tensions in planes at angles of 45° to the direction of those forces, and at right angles to one another amounts in magnitude to √2.
(5) A strain compounded of a simple longitudinal extension x, and a simple longitudinal condensation of equal absolute value, in a direction perpendicular to it, is a strain of magnitude x √2j or, which is the same thing (if σ = 2z), a simple distortion such that the relative motion of two planes at unit distances parallel to either of the planes bisecting the angles between the two planes mentioned above is a motion σ parallel to themselves, is a strain amounting in magnitude to -Î- .
5/ -
(6) If a strain be such that a sphere of unit radius in the body becomes an ellipsoid whose equation is
(1 - A)X 2 + (1 - B)Y 2 + (1 - C)Z 2 - DYZ - EZX - FXY = 1, the values of the component strains corresponding, as explained in Example (2 of Chap. IX., to the different coefficients respectively ,>re
For the components corresponding to A, B, C are simple longitudinal strains, in which diameters of the sphere along the axes of coordinates become elongated from 2to2 + A,2⅛B, 2 + C respectively: D is a distortion in which diameters in the plane YOZ, bisecting the angles YOZ and ΥΌΖ, become respectively elongated and contracted from 2 to 2 -¼ ⅛D, and from 2 to 2 - JD; and so for the others. Hence, if we take x, y, a, ζ, η, ξ to denote the magnitudes of six component strains, according to the orthogonal system of types described in Examples (1) and (2) of Chap. IX., the resultant strain equivalent to them will be one in whieh a sphere of radius 1 in the solid becomes an ellipsoid whose equation is
(1 - 2x)X 2 + (1 - 2⅛j Y 2 + (1 - 2i)Z2 - 2 √2(f YZ + qZX + ξXY) = 1, and its magnitude will be
√ (x 2 + y 2 +-- 2 + ξ 2 + >(2 + ζ 2 ).
(7) The specifications, according to the system of reference used in the preceding Example, of the unit strains of the six orthogonal types defined in Example f 3) of Chap. IX. are respectively as follows :— [table]
where!, m, η, Γ, m', n' denote quantities fulfilling the following conditions:—
Z 2 +m 2 ⅛n 2 =1, I +m +» =0, H' +mm'-^-nn ’ - 0, F 2 +m' 2 +n' 2 = l, Γ +nι' ⅛n' =0.
(8) If (1 -2eP)X 2 +(l-2cQ)Y' 2 +(l-2«R)Z 2 —2e√2(SYZ+TZX⅛L τ XY) = 1
De the equation of the ellipsoid representing a certain stress, the amount of work done by this stress, if applied to a body while acquiring the strain represented by the equation in the preceding example, will be
Px 4- Qy + 4-Tη 4- U£.
Cor. ∏ence, if variables X, Y, Z be transformed to any other set (X', Y', 7. ’ ) fulfilling the condition of being the coordinates of the same point, referred to another system of rectangular axes, the coefficients x, y, z, &c., x z , y z , z„ &e., in two homogeneous quadratic functions of three variables,
(1 - 2x)X 2 + (1 - 2y)Y 2 + (1 - 2z)Z 2 - 2 √2(ξYZ + qZX ⅛ ζXY) and
(1 - 2x,)X 2 + (1 - 2y z )Y a + (1 - 2z,)Z 2 - 2 √2(ξ,YZ⅛ q i ZX⅛ ζ,XY) , and the corresponding coefficients x', y', z', <fcc., x ’ z , y ’ ( , z ’ ; , <fcc., in these functions transformed to x ’, y ’, z ’, will be so related that x'x ’ z + y'√ z + z'∕ y + ζ'ζ' ι + η ' n ' l ⅛ ζ ’ ζ',=xx,+yy, +zt l + ζζ,+ ’Pl, + <X; or the function xx,÷y,V, + zz,+ of the coefficients is an “invariant”
for linear transformations fulfilling the conditions of transformation from one to another set of rectangular axes, since x⅛y+z and x,+y,÷s z are clearly invariants also, it follows that AA,+BB,+CC, + 2Dt>,-∣-2EE,-ι-2FF, is an invariant function of the coefficients of the two quadratics
AX 2 +BY 2 +CZ 2 +2DYZ+2EZX+2FXY r and A,X 2 +B,Y 2 -∣-C,Z 2 +2D∕YZ+2E,ZX+2F,XY, which it is easily proved to be by direct transformation.
This is the simplest form of the algebraic theorem of invariance with which we are concerned.
Chapter XI.— On Imperfect Concurrences of two Stress or Strain Types. Def. The concurrence of any stresses or strains of two stated types is the proportion which the work done when a body of unit volume experiences a stress of either type, while acquiring a strain of the other, bears to the product of the numbers measuring the stress and strain respectively.
6*or. 1. In orthogonal resolution of a stress or strain, its component of any stated type is equal to its own amount multiplied by its concurrence with that type; or the stress or strain of a stated type which, along with another or others orthogonal to it, have a given stress or strain for their resultant, is equal to the amount of the given stress or strain reduced in the ratio of its concurrence with that stated type.
Cor. 2. The concurrence of two coincident stresses or strains is unity; or a perfect concurrence is numerically equal to unity.
Cor. 3. The concurrence of two orthogonal stresses and strains is zero.
Cor. 4. The concurrence of two directly opposite stresses or strains is -1.
Cor. 5. If as, y, z, ξ, η, (?, are orthogonal components of any strain or stress r, its concurrences with the types of reference are respectively
£. V, £ δ *1 1
r ’ r r ’ r ’ r ’ r∙ ’ where r = (x 1 + y ’-i-z i + ζ ’ + q i + ξ ’) 1.
Cor. 6. The mutual concurrence of two stresses or strains is ll' + mm' + nn' + λλ' + uμ' + υυ , if l, m, η. λ, μ, V denote the concurrences of one of them with six orthogonal types of reference, and Γ, m', n' , λ', √, v ’ those of the other.
Cor. 7. The most convenient specification of a type for strains or stresses, being in general a statement of the components, according to the types of reference, of a unit strain or stress of the type to be specified, becomes a statement of its concurrences with the types of reference when these are orthogonal.
Examples— (1) The mutual concurrence of two simple longitudinal strains or stresses, inclined to one another at an angle 0, is cos 2 0.
(2) The mutual concurrence of two simple distortions in the sane plane, whose axes are inclined at an angle 0 to one another, is cos® 0 - sin 2 0, or 2 sin (45°- 6) cos (45°- 0).
Hence the components of a simple distortion 0 along two rectangular axes in its plane, and two others bisecting the angle between these taken as axes of component simple distortions, are
<5 (cos 2 0 — sin 2 0) and δ.2 sin 0 cos 0
respectively, if 0 be the angle between the axis of elongation in the given distortion and in the first component type.
(3) The mutual concurrence of a simple longitudinal strain and a simple distortion is
√2.cos a cos β, if a and β be the angles at which the direction of the longitudinal strain is inclined to the line bisecting the angles between the axes of the distortion; it is also equal to ⅛> ( cos2 Φ ~ c0s2 ψ) »
if φ and ψ denote the angles at which the direction of the longitudinal strain is inclined to the axis of the distortion.
(4) The mutual concurrence of a simple longitudinal strain and of a uniform dilatation is —77
√3
(5) The specifying elements exhibited in Example (7) of the preceding Chapter are the concurrences of the new system of orthogonal types described in Example (3) of Chap. IX. with the ordinary system, Examples (1) and (2), Chap. IX.
Chapter XII.— On the Transformation of Types of Reference for Stresses or Strains.
To transform the specification (at, y, z, ξ, η, ζ) of a stress or strain with reference to one system of types into (at 1 , a⅛ a⅛, κ 4 , x 5 , x 6 ) with reference to another system of types. Let (α j , b 1 , c 1 , e l , f l , g 1 ) be the components, according to the original system, of a unit strain of the first type of the new system; let (α 2 , b. 2 , c 2 , e 2 , ∕ 2 , g a ) be the corresponding specification of the second type of the^ new system; and so 011. Then we have, for the required formulae of transformation—
X = α 1 x 1 +α 2 x 2 +α 3 x 3 +α 4 x 4 +α s x 5 +α 6 x 6 , J∕ = δ l X 1 + ⅛T2 + f*3∙r3. + ¼∙' ,t '4+^i' r 5 +⅛ 2 ^β > Example.— The transforming equations to pass from a specification (x, y, ,, ξ, 1;, ζ) in terms of the system of reference used in Examples (C) and (7), Chapter X., to a specification (σ, ξ, η, ξ, w, ω) in terms of the new system described in Example (3) of Chap!eι IX., and specified in Example (7) of Chapter X., are as follows :—
1
x≈= y-σ + ⅛ + ⅛
1 y = ω
1
i=ξy,j σ + 7i3r τ n α, ,
W- »i-n, ξ=ζ; where, as before stated, I, m, η, V, m', n' are by quantities fulfilling the conditions Z 2 -j-m 2 4-n 2 =1, f +?n +∏ =0, i ' 2 +m' 2 +n ,2 =l r+m' +n'=0, H'4-nιm'+∏n'=0, Part II.— On the Dynamical Relations between Stresses and Strains experienced by an Elastic Solid. Chapter XIII.— Interpretation of the Differential Egnation of Energy.
In a paper on the Thermo-elastic Properties of Matter, published in the first number of the Quarterly Mathematical Journal, April 1855, and republished in the Plcilosophical Magazine, 1877, second half year, it was proved, from general principles in the theory of the Transformation of Energy, that the amount of work (w) required to reduce an elastic solid, kept at a constant temperature, from one stated condition of internal strain to another depends solely on these two conditions, and not at all on the cycle of varied states through which the body may have been made to pass in effecting the change, provided always there has been no failure in the elasticity under any of the strains it has experienced. Thus fora homogeneous solid homogeneously strained,it appears that w is a function of six independent variables x, y, z, ξ, η, ζ, by which the condition of the solid as to strain is specified. Hence to strain the body to the infinitely small extent expressed by the variation from (x, y, z, ξ,η, Øto (≈ + c ⅛ y + dy,z + dz, ξ + dξ, n + dη, ζ+dζh the work required to be done upon it is
⅛ + ¾, + ‰+⅛ + ⅛ + i⅜.
dx dy dz dζ dη α S>
The stress which must be applied to its surface to keep the body in equilibrium in the state (.υ, y, z, ξ, η, ζ} must therefore be such that it would do this amount of work if the body, under its action, were to acquire the arbitrary strain dx, dy, dz, dξ, dη, dζ ; that is, it must be the resultant of six stresses:—one orthogonal to the five strains dy, dz, dξ, dη, dζ, and of such a magnitude as to do the work dx when the body acquires the strain dx , ; a second orthogonal to dx, dz, dξ, dη, dζ, and of such a magnitude as to do the work dy when the body acquires the strain dy ; and so on. If a, b, c, f, g, h denote the respective concurrences of these six stresses, with the types of reference used in the specification (.τ, y, z, ξ, ∙η, ζ) of the strains, the amounts of the six stresses which fulfil those conditions will (Chapter XI.) be given by the equations
P_1 dw Q = ll R=l^f ä dx' b dy ’ c dz'
1 dw 1 dw ττ 1 dw, s ~∕dξ τ =jd η ' U_ A dζ'
and the types of these component stresses are determined by being orthogonal to the lives of the six strain-types, wanting the first, the second, &c., respectively.
Cor. If the types of reference used in expressing the strain of the body constitute an orthogonal system, the types of the component stresses will coincide with them, and each of the concurrences will be unity. Hence the equations of equilibrium of an elastic solid referred to six orthogonal types are simply dw dw _ dw V = dx' ci ~dy' ^~jdz dw dw „ w dξ d η <Zξ
Chapter XIV.— Reduction of the Potential Function, and of the Eqzιations of Equilibrium, of an Elastic Solid to their simplest Forms.
If the condition of the body from which the work denoted by w is reckoned be that of equilibrium under no stress from without, and if x, y, z, ξ, η, ζ be chosen each zero for this condition, we shall have, by Maclaurin’s theorem, w=H 2 (αr, y, z, ζ, η, ζ) + ∏ 3 (z, y, z, ξ, η, ζ) + <fcc., where H 2 , II 3 , &c., denote homogeneous functions of the second order, third order, &c., respectively. Hence f yy ’ &c., will each be a linear function of the strain coordinates, together with functions of higher orders derived from H 3 , &c. But experience shows (section 37 above) that, within the elastic limits, the stresses are very nearly if not quite proportional to the strains they are capable of producing; and therefore H 3 , &c., maybe neglected, and we have simply w=H 2 (,r, y,z, ζ, η, ζ ).
Now in general there will be twenty-one terms, with independent coefficients, in this function; but by a choice of types of reference, that is, by a linear transformation of the independent variables, we may, in an infinite variety of ways, reduce it to the form w=l 2 (Az 2 + B ÿ 2 + Cs 2 +Fξ≡ + G ij 2 + Hξ2).
The equations of equilibrium then become
P = -Λ∙, Q = -y, R = S, a b c∙ s= r -∈, τ=‰, u=⅛
∕ ÿ Λ
the simplest possible form under which they can be presented. The interpretation can be expressed as follows.
Prop. An infinite number of systems of six types of strains or stresses exist in any given elastic solid such that, if a strain of any one of those types be impressed on the body, the elastic reaction is balanced by a stress orthogonal to the five others of the same system.
C , haγter XV.— Ou the Six Principal Strains of an Elastic Solid. To reduce the twenty-one coefficients of the quadratic terms in the expression for the potential energy to six by a linear transformation we have only fifteen equations to. satisfy; while we nave thirty disposable transforming coefficients, there being five independent elements to specify a type, and six types to be changed. Any further condition expressible by just fifteen independent equations may be satisfied, and makes the transformation determinate. Now the condition that six strains may be mutually orthogonal is expressible by just as many equations as there aie diilerent pairs of six things, that is, fifteen. The well-known algebraic theory of the linear transformation ot quadratic functions shows for the case of six variables—(1) that the six coefficients in the reduced form are the roots of a “determinant” of the sixth degree necessarily real; (2) that this multiplicity of roots leads determinately to one, and only one system of six types fulfilling the prescribed conditions, unless two or more of the roots are equal to one another, when there will be an infinite number of solutions and definite degrees of isotropy among them; and (3) that there is no equality between any of the six roots of the determinant in general, when there are twenty-one independent coefficients in the given quadratic.
Prop. Hence a single system of six mutually orthogonal types may be determined for any homogeneous elastic solid, so that its potential energy when homogeneously strained in any way is expressed by the sum of the products of the squares of the components of the strain, according to those types, respectively multiplied by six determinate coefficients.
Def. The six strain-types thus determined are called the Six # Principal Strain-types of the body.
The concurrences of the stress-components used in interpreting the differential equation of energy with the types of the straincoordinates in terms of which the potential of elasticity is expressed, being perfect when these constitute an orthogonal system, each of the quantities denoted above by a, b, c, f, g, h, is unity when the six principal strain-types are chosen for the coordinates. The equations of equilibrium of an elastic solid may therefore be expressed as follows :—
P-=Aατ, Q = By, R = C⅛
S-Fξ, T=Gιj, U = Hζ, where x, y, z, ξ, ∙η, ζ denote strains belonging to the six Principal Types, and P, Q, R, S, T, U the components according to the same types, of the stress required to hold the body in equilibrium when in the condition of having those strains. The amount of work that must be spent upon it per unit of its volume, to bring it to this state from an unconstrained condition, is given by the equation w=∣(Aχ2 + Bj/2 + Cz» ÷ Fξ β + G> i 2 + Hζ 2 ).
Def. The coefficients A, B, C, F, G, H are called the six Principal Elasticities of the body.
The equations of equilibrium express the following propositions :—
Prop. If a body be strained according to any one of its six Principal Types, the stress required to hold it so is directly concurrent with the strain.
Examples.— (1) If a solid be cubieally isotropic in its elastic properties, as crystals of the cubical class probably are, any portion of it will, when subject to a uniform positive or negative normal pressure all round its surface, experience a uniform condensation or dilation in all directions. Hence a uniform condensation is one of its six principal strains. Tiιree plane distortions with axes bisecting the angles between the edges of the cube of symmetry are clearly also principal strains, and since the three corresponding principal elasticities are equal to one another, any strain whatever compounded of these three is a principal strain. Lastly, a plane distortion whose axes coincide with any two edges of the cube, being clearly a principal distortion, and the principal elasticities corresponding to the three distortions of this kind being equal to one another, any distortion compounded of them is also a principal distortion.
Hence the system of orthogonal types treated of in Examples (3) Chap. IX., und (7) Chap. X., or any system in which, for (II.), (III.), and (IV.), any three orthogonal strains compounded of them are substituted, constitutes a system of six Principal Strains in a solid cubieally isotropic. There are only three distinct Principal Elasticities for such a body, and these are—(A) its modulus of compressibility, (B) its rigidity against diagonal distortion in any of its principal planes (three equal elasticities), and (C) its rigidity against rectangular distortions of a cube of symmetry (two equal elasticities).
(2) In a perfectly isotropic solid, the rigidity against all distortions is equal. Hence the rigidity (B) against diagonal distortion must be equal to the rigidity (C) against rectangular distortion, in a cube; and it is easily seen that if this condition is fulfilled for one set of three rectangular planes for which a substance is isotropic, the isotropy must be complete. The conditions of perfect or spherical isotropy are therefore expressed in terms of the conditions referred to in the preceding example, with the farther condition B = C.
A uniform condensation in all directions, and any system whatever of five orthogonal distortions, constitute a system of six Principal strains in a spherically isotropie solid. Its Principal Elasticities are simply its Modulus of Compressibility and its Rigidity.
Prop. Unless some of the six Principal Elasticities be equal to one another, the stress required to keep the body strained otherwise than according to one or other of six distinct types is oblique to the strain.
Prop. The stress required to maintain a given amount of strain is a maximum or a maximum-minimum, or a minimum, if it is of one of the six Principal Types.
Cor. If A be the greatest and H the least of the six quantities A, B, C, F, G, H, the principal type to which the first corresponds is that of a strain requiring a greater stress to maintain it than any other strain of equal amount; and the principal type to which the last corresponds is that of a strain which is maintained by a lest stress than any other strain of equal amount in the same body. The stresses corresponding to the foftr other principal strain-types have each the maximum-minimum property in a determinate way.
Prop. If a body be strained in the direction of which the concur- . rences with the principal strain-types are l, m, n, κ, μ, v, and to an amount equal to r, the stress required to maintain it in this state will be equal to ∩r, where
Ω = (A»Z* + B W + CM + F i ∕V+G 2 μ 2 ⅛ II≡ i y≡)i , and will be of a type of which the concurrences with the principal types are respectively
AZ Bzn Cn Fλ Gμ H∣∕ Q’ Ω“’ Ω^ ’ Ω^' Q^ ’ Ω^ ’
Prop. A homogeneous elastic solid, crystalline or non-crystalline, subject to magnetic force or free from magnetic force, has neither any right-handed or left-handed, nor any dipolar, properties dependent on elastic forces simply proportional to strains.
Cor. The elastic forces concerned in the luminiferous vibrations of a solid or fluid medium possessing the rightor left-handed property, whether axial or rotatory, such as quartz crystal, or tartaric acid, or solution of sugar, either depend on the heterogeneousness or on the magnitude of the strains experienced.
Hence as they do not depend ou the magnitude of the strain, they do depend on its heterogeneousness through the portion of a medium containing a wave. Cor. There cannot possibly be any characteristic of elastic forces simply proportional to the strains in a homogeneous body, corresponding to certain peculiarities of crystalline form which have been observed,—for instance corresponding to the plagiedral faces discovered by Sir John Herschel to indicate the optical character, whether right-handed or left-handed, in different specimens of quartz crystal, or corresponding to the distinguishing characteristics of the crystals of the right-handed and left-handed tartaric acids obtained by Μ. Pasteur from racemic acid, or corresponding to the dipolar characteristics of form said to have been discovered in electric crystals.
Chapter XVI.— Application of Conclusions to Natural Crystals.
It is easy to demonstrate that a body, homogeneous when regarded on a large scale, may be constructed to have twenty-one arbitrarily prescribed values for the coefficients in the expression for its potential energy in terms of any prescribed system of strain coordinates. This proposition was first enunciated in the paper on the Thermo-elastic Properties of Solids, published April 1855, in the Quarterly Mathematical Journal alluded to above. We may infer the following.
Prop. A solid may be constructed to have arbitrarily prescribed values for its six Principal Elasticities and an arbitrary orthogonal system of six strain-types, specified by fifteen independent elements, for its principal strains: for instance, five arbitrarily chosen systems of three rectangular axes, for the normal axes of five of the Principal Types; those of the sixth consequently in general distinct from all the others, and determinate; and the six times two ratios between the three stresses or strains of each type, also determinate. The fifteen equations expressing (Chap. VI.) the mutual orthogonality of the six types determine the twelve ratios for the six types, and the three quantities specifying the axes of the sixth type in the particular case here suggested: or generally the fifteen equations determine fifteen out of the thirty quantities (viz. twelve ratios and eighteen angular coordinates) specifying six Principal Types.
Cor. There is no reason for believing that natural crystals do not exist for which there are six unequal Principal Elasticities, and six distinct strain-types for which the three normal axes constitute six distinct sets of three principal rectangular axes of elasticity.
It is easy to give arbitrary illustrative examples regarding Principal Elasticities: also, to investigate the principal strain-types and the equations of elastic force referred to them or to other natural types, for a body possessing the kind of symmetry as to elastic forces that is possessed by a crystal of Iceland spar, or by a crystal of the “tesseral class,” or of the included “cubical class.” Such illustrations and developments, though proper for a students’ text book of the subject, are unnecessary here.
For applications of the Mathematical Theory of Elasticity to the question of the earth’s rigidity and elasticity as a whole, and to the equilibrium of elastic solids in general, which are beyond the scope of the present article, the reader is referred to Thomson and Tait’s Natural Philosophy, §§ 588, 740, 832, 849, and Appendix C.
Chapter XVII.— Plane Waves in a Homogeneous Aeolotropic Solid. A plane wave in a homogeneous elastic solid is a motion in which every line of particles in a plane parallel to one fixed plane experiences simply a motion of translation—but a motion differing from the motions of particles in planes parallel to the same. Let OX, OY, OZ be three fixed rectaιlgular axes; OX perpendicular to the wave front (as any of the parallel planes of moving particles referred to in the definition is called), and OY, OZ in the wave front. Let x + u, y + v, z + w be the coordinates at time t of a particle which, if the solid were free from strain, would be at (x, y, z). The definition of wave motion amounts simply to this, that u, v, w are functions of x and t.
The strain of the solid (Chap. VII. above) is the resultant of a simple longitudinal strain in the direction OX, equal to and dv die two differential slips ’ parallel to OY and OZ, constituting simple distortions of which the numerical magnitudes (Chap. X.) are w* χ 2, and y β V2.
dx dx
Γut then du c dv , dw , y -r-'-=c< -√2=ιj, — √2 = ζ .... (1>:
dx dx dx b and let W denote the work per unit of bulk required to produce the strain represented by this notation. We have (Chap. XV. )
W = i(A^-t-B 1 j 2 + Cζ 2 + 2Dηζ + 2Eζ∈ + 2Ff,j) . . . (2).
where A, B, C, D, E, F denote moduluses of elasticity of the solid. Let ρ, ' q, r denote the three components of the traction per unit area of the wave front. We have (Chap. XV.)
p = Aξ+Fη + Eζ)
0v z l = Ff+B ’ ∕ + I}⅛ > (3), r√∣ = Eξ 4- D>j + Cζ )
Now let ξ, η, ζ be taken such that
Aξ+F,) + Eζ=Mn
Fξ + B f ) + Dζ = i VI l A (4)
Eξ + D,∣ + Cζ=Mζj the determinantal cubic gives three real positive values for M, and with M equal to any one of these values, (4) determine the ratios ξ .η ∙. ζ. Hence when the solid is strained in any one of the three ways thus determined we have
.,<Zu Λtdv ..die . ,
J> = M-, ?=M—, r = M- .... (5).
dx dx dx
The three components of the whole force due to the tractions on the sides of an infinitely small parallelepiped δx, δy, δs of the solid are clearly * ’ Tδχ.δyδz, ^lδx.δyδz, and clr δx.δyδz . . . . (6), dx dx dx ,, and therefore, if p be its density, and consequently pδx δy δz its mean, the equations of its motion are d 2 uifdp d 2 υ dq d 2 w _dr Pdi 2 ~dx P dt 2 ~dx ’ Pdx 2 ~dx - (7) ·
These, putting for p, q, r their values by (5), become d 2 u .,d 2 u d 2 v . r d 2 v d 2 u> .,d 2 u> PdΓ 2 ~ yl dx 2 ' Pdl 2 ~^dx 2 ' Pdt 2 ~ yl dx 2 (8) ·
And by (4) and (1) we have
Au-i-(Fυ-∣-Ew)√2 = Mu "∣
Fu +(Bι>+Dκ∙)√2 = Mr√2 > (9).
Eu +(Dp∙ ∣ -Cw)√2=Mw√2 J
Let M 1 , M 2 , M 3 be the three roots of the determinantal cubic, and b 1 , c 1 ∙, b i , c i ∙, b 3 , c 3 , the corresponding values of the ratios -, - determined by (9). The complete solution of (8), subject to
(9), is u =u 1 +u 2 +u 3 V =διUι4-Z ’ 2 u 2*i"^3 w 3 1
W =CιUι-t-C 2 U24^ f 3 u 3 I
where u 1 = ∕ 1 (i , +t√— 1 j + F 1 (r-(√-j
∖ ι 0 , × P , ) (10), u 2 =∕ 2 (x + i√^) + F 2 (x-i√½) 1
V3=√i(*+<√^~)+ 1 '3(iΓ* v 'y) ∕
l 1 , ?ι,Λ, F,, ∕ 3 , F 3 denoting arbitary functions. Hence we conclude that there are three different wave-velocities,
√^1, √ i b v ½ P P P and three different modes of waves, determined by equations (9).
Waves in an Isotropic Solid. — If the solid be isotropic, we have
B = C )
d=e≈=f=o y ... . (il)
M,≡>A. M, = M,-=≈B I Hence, instead of three different waves with different velocities, we have just two,—a wave (like that of sound in air or other elastic fluid) in which the motions are perpendicular to the wave front, and the other (like the waves of light in an isotropic medium) in which the motions are parallel to the wave front.
Waves in an Incompressible Solid (Aeolotrognc or Isotropic).— If the solid be incompressible, we have A= oo, and u must be zero.
Hence w=B√ ’ +cζ*+2D ll ζ
and by a determinantal quadratic, instead of cubic, we find two wave-velocities and two wave-modes, in each of which the motion is parallel to the wave front. In the case of isotropy the two wave velocities are equal.
It is to be noticed that M 1 , M 2 , M 3 in the preceding investigation are not generally true “principal moduluses,” but special moduluses corresponding to the particular plane chosen for the wave front. In the particular case of isotropy, however, the equal moduluses M 2 , M 3 of (11) are principal moduluses, being each equal to the modulus of rigidity, but M 1 is a mixed modulus of compressibility and rigidity—not a principal modulus. In the case of irfcompressi-bility, the two moduluses found from the determinantal quadratic by the process indicated above are not principal moduluses generally, because the distortions by the differential motions of planes of particles parallel to the wave front must generally give rise to tangential stresses orthogonal to them, which do not influence the wave motion.
