Elasticity, the property in virtue of which a body tends to recover its form and dimensions on the removal of the forces by which these have been changed. A perfectly elastic body is defined by Thomson and Tait (1872) as one which " when brought to any one state of strain requires at all times the same stress to hold it in this state, however long it may be kept strained, or however rapidly its state may be altered from any other strain or from no strain to the strain in question." According to Maxwell (1872), a perfectly elastic body is one "which, subjected to a given stress at a given temperature, experiences a strain of definite amount, which does not increase when the stress is prolonged, and which disappears completely when the stress is removed." If the form of the body is found to be permanently altered, its state of stress just previous to the change is called the limit of perfect elasticity. If the stress increases until the body breaks, the value of the stress is called the strength of the material. If breaking takes place before there is any permanent alteration of form, the body is said to be brittle. If the stress, when it is maintained constant, causes a displacement within the body which increases continually with the time, the substance is said to be viscous.
Viscosity, whether in solids or fluids, is then intimately connected with their elasticity, and the preceding definitions of elasticity, although popularly associated with the behavior of solid bodies, are equally applicable to liquids and gases; indeed, strictly interpreted, they have reference to the behavior of the molecules and ultimate atoms of which bodies are supposed to be composed. A perfect knowledge of the laws of elasticity is nothing less than a knowledge of the exact laws according to which atoms attract and repel each other, or, more generally, of the ultimate constitution of matter. In this broad sense this subject offers the most important and at the same time the most difficult field of study that can be cultivated by the physicist. The ultimate constitution of matter is in fact as truly the goal of the modern physical and chemical sciences, as the constitution of the universe is the final problem of astronomy. The tendency of modern physics is to look with distrust upon all theories of molecular action that assume the principle of action at a distance. When, however, we have attained to a correct knowledge of the internal constitution of bodies, it may be expected that the laws of elasticity will be deduced therefrom by mathematical processes.
Until then the subject of our article must be treated inductively, except in so far as we assume the relative position of the particles of bodies to be disturbed only to an indefinitely small degree by the exterior forces; in these cases the molecular motions may be deduced from the established general laws of mechanics, and have indeed been so investigated with great success by a host of scientists, beginning with Bernoulli, Young, Fresnel, and Green. The general investigation of the relation between the strains and deformations of a solid has been shown by Green (1830) to depend upon the solution of a quadratic equation having 6 unknown quantities and 21 terms whose coefficients are essential for a complete theory of the dynamics of an elastic solid subjected to infinitely small strains. I. Elasticity of Masses. The elastic properties of homogeneous bodies relate to the behavior either of the constituent molecules and sensible masses of the bodies or of their ultimate atoms to each other. In considering the latter class of relations we have to do with chemical changes and the properties of heat, light, electricity, and magnetism. Considering the behavior of molecules, we have to do with the laws of strength of materials, elasticity of springs, propagation of sound, etc.
The latter class of phenomena will here first attract our attention, and we shall treat in succession of solids, jellies, fluids, vapors, and gases, concluding with some of the relations between the molecular elasticities of these bodies and the agencies of heat, light, and electricity. - Our first ideas as to the elasticity of solids are derived from the properties of extension and compression under forces respectively of tension and pressure. In the second column of the following table is given for each substance the weight necessary to extend by 1/100 part of its length (or one centimetre) a bar whose section is one square millimetre, and whose length is one metre:
Weight In kilogrammes.
Modulus in kilogrammes per Bq. mm.
Modulus in pounds per sq. inch.
Modulus of length in metres.
When the elongations are small they are, according to the law of Hooke, directly proportioned to the tension or the applied weight; therefore in the preceding cases weights ten times as great as those in the second column will extend the bars to double their original length, if we suppose the material to withstand such a force without breaking; the corresponding numbers are found in the third column of the preceding table, and constitute for each substance the so-called "Young's modulus of elasticity," which may be defined as the weight that will so. elongate a bar of unit square section as to double its original length. The moduli given in the third column in kilogrammes per square millimetre may be converted into the British system of pounds per square inch by multiplying by 1,422.3, as given in the fourth column. It is frequently convenient to express the above given weight modulus by an equivalent length of a bar of the same material; this number, which may be called the length modulus, is found by dividing the weight modulus by the weight of the unit of length, or in the French system, by the specific gravity of the substance and multiplying the quotient by 100,000. Thus a bar of glass will be stretched to double its length by the weight of a similar bar of glass whose length is 19,000 metres.
In general, for large forces and changes of form that leave a permanent deformation of the solid, the resistance to crushing differs from the resistance to stretching; this is shown by the coefficients of strength for these two kinds of forces; and the difference of bodies in this respect is a highly important element in calculating their fitness for building or mechanical purposes. On the other hand, the effect of an exceedingly small force in producing a slight compression is sensibly equal to the effect of the same force in producing a slight extension; therefore the above given moduli of elasticity hold good for such values of both pressure and tension as do not approach the limit of perfect elasticity. If now a bar of any substance be slightly bent, its convex side is extended, but its concave side compressed; there must therefore be within its substance some neutral line, or axis of no change whatever; this line constitutes the elastic curve of Bernoulli, and has been the subject of many investigations. The exact curvature of this line, or of the bar, is known when the applied forces are given.
The formulas relating to the various cases that occur in engineering are given in works on applied mechanics, such as those of Weisbach and Rankine. One of the most interesting and important applications of the elasticity of a plane curved spring is found in the case of the mainspring and especially of the hair spring of a chronometer; the theory of their action has been studied, among others, by Yvon Villarceau (18G3). Other applications are found in the dynamometer for measuring great tensions, the horizontal fixed glass thread for weighing minute quantities, the tuning fork and the stretched strings of musical instruments, etc. - The near approach to perfect elasticity that is the property of some bodies, as steel and glass, is evinced by the perfect uniformity of the times of vibration of the strings, bells, bars, etc, that give out pure musical notes on being struck, and that no matter how extensive their arcs of vibration or how loud their corresponding notes; if the elastic force did not increase precisely in proportion to the amount of the molecular displacement, we should notice discords instead of the single pitch that emanates from the sounding body.
The perfect uniformity of these minute vibrations has suggested their application in the sphygmograph and other chronographs, where they replace the pendulum for the measurement of minute intervals of time; indeed, in the chronograph of M. Hipp, a vibrating spring serves also as the regulator of the revolving cylinder. The vibrations of a carefully arranged metallic spring, hermetically sealed in an exhausted glass vessel, and at a standard temperature, depending as they do solely on the elasticity of the body, offer a means of measuring small intervals of time with almost absolute perfection; and Sir William Thomson has suggested that for scientific purposes such an arrangement must be adopted as a standard far more constant than the rotation of the earth or the vibration of a pendulum. - If a straight bar be twisted about its longitudinal axis, an elastic force is brought into play to resist the mechanical couple by which the twisting is effected. This, the elasticity of torsion, does not depend upon that of compression or extension; its modulus must therefore be determined by direct experiment.
The laws of the elasticity of torsion have been studied originally by Coulomb, and subsequently by Binet, J. Thomson, and especially by St. Venant (1855) and Thomson and Tait. The principal applications of torsional elasticity are found in the spring balance in ordinary commercial use, and in the torsion balance used in electrical measurements, in the Cavendish experiment, and other delicate researches. In the latter instrument the amount of torsion is evidently a direct measure of the external force applied and to be measured. In the former instrument (the spring balance) the elastic wire is coiled around a cylinder like the threads of a screw, and an external force is applied parallel to the axis of the cylinder to increase or diminish the distance between the coils. The theory of the action of such spiral or helicoidal springs has been developed by Binet (1814), and by Prof. J. Thomson (1848), by whom it has been shown that the force that opposes the elongation or contraction of a helicoidal spring is the elasticity of torsion.
Besides the applications above mentioned of the spring balance to commercial uses, it has been proposed to use a very delicate i instrument of this construction as a means of investigating the local and general variations of gravity on the earth's surface, for which purpose it possesses some advantages over the pendulum. - The grandest application of the laws of the elasticity of solids that has yet been made consists in the investigation into the elasticity of the earth, considered as a whole, which is an important element in the theory of oceanic tides. Sir William Thomson, who has given much attention to this problem, concludes (1871) that the earth has considerably more average rigidity than a globe of glass of the same size; were it not so, the yielding of the solid earth to the tidal influence of the sun and moon would to a great degree annul the observed ocean tides. - The elasticity of a body subjected to a severe strain is in general permanently injured; and Messrs. Hodgkinson and Fairbairn (1837) concluded from their experiments on iron, etc, that the limit of perfect elasticity is much lower than was formerly conceived; that, indeed, a permanent set takes place in ordinary cast iron on the application of even a very slight force; and that there is in general no clearly defined limit of perfect elasticity.
This conclusion is probably applicable only to the metals as found in commerce, and not to chemically pure homogeneous or crystalline masses. The ease with which a solid body receives a slight permanent set or deformation is expressed by the words soft and plastic, and the stress on a unit section exerted at the time the body breaks or crushes is the coefficient of rupture; at the limit of perfect elasticity the investigation of elasticity proper ceases, and the phenomena become those of viscosity, as previously defined, until the force applied causes a rupture of the body. - In connection with the elastic properties of solids there remain to be noticed the phenomena of impact. When two bodies strike each other, the portions in contact are forcibly compressed; and as no body is perfectly inelastic, they begin to separate as soon as the compression reaches its maximum value. Newton found that the relative velocity of separation after impact bears a proportion to the previous relative velocity of approach, which is constant for the same two bodies; this proportion is always less than unity, but approaches that limit the harder the bodies are.
To this proportion the name coefficient of elasticity is frequently given; this, however, is a misnomer, and Thomson and Tait suggest the more appropriate term coefficient of restitution, while others call it the coefficient of impact. The quantity of force, if any, that is apparently annulled on the collision of two bodies, is not destroyed, but is converted into the other forms of molecular elasticity, i. e., the vibrations of sound, heat, light, and actinism, which spread in all directions to indefinite distances from the place of their origin. - Crystalline bodies differ from homogeneous ones, so far as concerns their elastic properties, only in that the coefficient of elasticity varies with the direction of the external forces that produce the strain. - Jellies form a class of solids distinguished by a wide range of distortion possible within the limits of perfect elasticity, or without producing permanent deformation. In solid bodies the limit of perfect elasticity varies from something indefinitely small, as in most hard bodies, to a number as large as 1.1 or 1.2, as in the case of cork, and to 2 or 3 in the case of India rubber; the latter is, in its purest state, more properly allied to the class of jellies.
In the vibration of solids it has been observed that a force generally called internal or molecular friction resists the vibrations and diminishes their amplitude; this resisting force is probably nothing but the transformation of a portion of the vibratory movement of the mass into heat vibrations. The development of internal friction is specially notable in the vibrations of jellies; it is allied to, if not the same as, the property of viscosity in fluids. - The distinctive property of a fluid is that its molecules can quickly, easily, and permanently change their relative positions; a perfect fluid is a body incapable of resisting a change of shape, provided only that its volume be not altered. Fluids are divided into the two classes of liquids and gases, according to the relative values of their coefficients of elasticity and viscosity. - The elasticity of liquids is generally known as their compressibility, which property has reference to the behavior of the interior of a mass of liquid; the phenomena of capillarity, on the other hand, depend on the peculiar elastic condition of the superficial film of a mass of liquid. "The coefficient of elasticity of a fluid is the ratio of any small increase of pressure to the cubical compression thereby produced.
The elasticity however varies with certain conditions that may be imposed, such as that the fluid be under a given pressure or exposed to a given temperature. Every substance has two elasticities, one corresponding to a constant temperature, the other corresponding to the case where no heat is allowed to escape from the body during the process of compression. In the latter case the elasticity is always greater than in the case of uniform temperature." The following are the coefficients of compressibility of several liquids, or the fractions by which the original volume is diminished for an increase of pressure of one atmosphere or about 15 lbs. to the square inch:
Coefficient of elasticity.
Mercury at a temperature of 32° Fahr
Water " " 32 "
" " " 107 "
Alcohol " " 45 "
Ether " " 32 "
" " " 57 "
The other and highly important property, the superficial elasticity of liquids, is commonly known as the phenomena of thin films, of capillarity, etc, and is well illustrated by the blowing of a soap bubble, which operation is but the forcible stretching of a superficial sheet of a highly elastic liquid membrane. The superficial tension of the latter may be measured in grammes weight per linear metre or other unit; it is the same whether the globe be hollow or full; it varies with different liquids and with the nature of the gas or other fluid contiguous to the thin film. The following table contains some results obtained by Quincke; the figures hold good for the temperature of 20° centigrade and the above units of weight and length :
Tension of surface separating the liquid from
The investigation of the superficial tension of liquids leads to the explanation of the phenomena of capillarity, evaporation, the wetting of surfaces, the condition of vapor in clouds, or of vapor vesicles if such there be, etc.; and thus brings these diverse subjects within the range of the laws of elasticity. - The elasticity of vapors, or more strictly their expansibility, is a matter of such importance in connection with the steam engine, etc, and in meteorology, that it will be specially treated of under those heads. By the removal of pressure or the communication of heat, or both", a solid or liquid may be converted into a more or less perfect gas, usually called a vapor. The vapors generally have coefficients of elasticity that depend almost wholly upon the temperature; a few such are given in the following table, from Regnault:
- The elasticity of a gas is its power to resist compression, and to restore either its original volume or its original tension. At ordinary temperatures all gases are perfectly elastic, so far as the most delicate measures have as yet been able to determine, for the most extreme changes of pressure. The limits of perfect elasticity are therefore practically infinite; thus justifying the law first announced by Boyle, though frequently spoken of as the law of Mariotte, because firmly established by him, that the density of a gas is directly, or its volume inversely, proportional to the pressure. The elasticity of gases, being to a high degree independent of temperature, finds an important application in mechanical engineering, in the transmission of power to great distances by means of compressed air. It is through this elastic property of the air that sounds are conveyed by it at the rate of about 1,100 feet per second in all directions. II. Elasticity of Molecules and Atoms. The explanation of the phenomena of heat and light has led to the assumption of the existence throughout all space of a highly elastic gas, the ether of physical science; the vibrations of the molecules of this ether, being communicated to the nerves of the body, produce the sensations of heat and light.
This gas is supposed to exist, though in a state of constraint, among the atoms of transparent bodies, and the mathematical development of the nature of its molecular vibrations offers a complete explanation of all the phenomena of optic and thermic science; and indeed, according to Maxwell and Edlund, we can use this same ether for the explanation of the phenomena of electricity and magnetism. Its elasticity is such that heat, light, and electricity are transmitted by it at the rate of about 200,000 miles per second through the interplanetary spaces. It is in general homogeneous, but in the interior of many crystalline bodies its elasticity varies as does that of the material of the crystal, thus producing the phenomena of polarization, double refraction, etc. While heat itself apparently consists of molecular vibrations depending on the elasticity of the ether, it on the other hand exerts a marked and highly important influence over the elasticity of bodies. So intimate is this connection that it may indeed be doubted whether the elasticity of the ether can be properly spoken of as distinct from the elasticity of the more material bodies that are recognized by the methods of chemical science.
In general an excess of heat induces such molecular vibrations that the constituent atoms become separated from each other, producing in the chemical nature of the body the changes known as dissociation, changes that may be compared with the rupture or crushing of a solid mass when subjected to too great a stress. When, on the other hand, a more moderate degree of heat is applied, most interesting changes occur, which will be briefly indicated in so far as they affect the elasticity of the substances. The elasticity of solid bodies is in general diminished by heat, thus tending to lengthen the time of vibration of chronometer springs, tuning forks, etc, retarding the velocity of sound waves, contracting the limits of perfect elasticity, diminishing the force of cohesion or the strength of the material, increasing its plasticity, diminishing its viscosity, etc. Iron and steel are among the few known instances of departure from this general rule. The elasticity of liquids is so affected by heat that, according to Gladstone and Dale, the index of refraction and the coefficient of dispersion diminish as the temperature increases. An increase of temperature diminishes the compressibility of water, but increases that of ether, alcohol, and chloroform.
With regard to vapors and gases there seems to be one uniform rule, i. e., that an increase of heat increases the elasticity of these forms of matter. The preceding table from Regnault has shown the connection between temperature and elastic pressure for some of the more interesting vapors. For short distances on either side of the boiling point of a liquid the elasticity of its vapor approximately varies directly as the temperature, as was first propounded by Dalton. In considering the effects of heat on the elasticity of a vapor or a gas, it is necessary to distinguish between (1) the elastic pressure exerted by a gas that is confined within a constant volume and has its temperature altered by external heat, and (2) the pressure exerted by a gas that is compressed by external force to a smaller volume, but whose temperature is not altered. The former is the elasticity under constant volume or the expansion under constant pressure ; the latter is the elasticity under constant temperature, and is for perfect gases expressed by Boyle's law, which however holds good only approximately for vapors.
Gay-Lussac has shown that the expansion under constant pressure is sensibly the same for all perfect gases, and is such that the volume increases by 0.002036 of its original value for an increase of 1° F., or by 0.00367 for an increase of 1° 0. This law was fully confirmed by Regnault, who showed that it is only approximately true when gases or vapors approach the point of liquefaction. The effect of heat on the elasticity of a gas is so combined with its effect on the density as in general to increase the velocity of the transmission of sound. Thus in air at a pressure of 30 inches the velocity in feet per second is 1,089 at a temperature of 32° F., but is 1,131 at a temperature of 70° F.