Page:EB1911 - Volume 09.djvu/163

depends upon the two constants $\lambda$ and $\mu .$ These two constants were introduced by G. Lamé in his treatise of 1852. The importance of the quantity $\mu$ had been previously emphasized by L. J. Vicat and G. G. Stokes.

27. The potential energy per unit of volume (often called the “resilience”) stored up in the body by the strain is equal to

${\tfrac {1}{2}}(\lambda +2\mu )(e_{xx}+e_{yy}+e_{zz})+{\tfrac {1}{2}}\mu (e_{yz}^{2}+e_{zx}^{2}+e_{zy}^{2}-4e_{yy}e_{zz}-4e_{zz}e_{xx}-4e_{xx}e_{yy}),$

or the equivalent expression

${\tfrac {1}{2}}[({\text{X}}_{x}^{2}+{\text{Y}}_{y}^{2}+{\text{Z}}_{z}^{2})-2\sigma ({\text{Y}}_{y}{\text{Z}}_{z}+{\text{Z}}_{z}{\text{X}}_{x}+{\text{X}}_{x}{\text{Y}}_{y})+2(1+\sigma )({\text{Y}}_{z}^{2}+{\text{Z}}_{x}^{2}+{\text{X}}_{x}^{2})]/{\text{E}}.$

The former of these expressions is called the “strain-energy-function.”

28. The Young’s modulus ${\text{E}}$ of a material is often determined experimentally by the direct method of the extensometer (§ 17), but more frequently it is determined indirectly by means of a result obtained in the theory of the flexure of a bar (see §§ 47, 53 below). The rigidity $\mu$ is usually determined indirectly by means of results obtained in the theory of the torsion of a bar (see §§ 41, 42 below). The modulus of compression $k$ may be determined directly by means of the piezometer, as was done by E. H. Amagat, or it may be determined indirectly by means of a result obtained in the theory of a tube under pressure, as was done by A. Mallock (see § 78 below). The value of Poisson’s ratio $\sigma$ is generally inferred from the relation connecting it with ${\text{E}}$ and $\mu$ or with ${\text{E}}$ and $k,$ but it may also be determined indirectly by means of a result obtained in the theory of the flexure of a bar (§ 47 below), as was done by M. A. Cornu and A. Mallock, or directly by a modification of the extensometer method, as has been done recently by J. Morrow.

29. The elasticity of a fluid is always expressed by means of a single quantity of the same kind as the modulus of compression of a solid body. To any increment of pressure, which is not too great, there corresponds a proportional cubical compression, and the amount of this compression for an increment $\delta p$ of pressure can be expressed as $\delta p/k.$ The quantity that is usually tabulated is the reciprocal of $k,$ and it is called the coefficient of compressibility. It is the amount of compression per unit increase of pressure. As a physical quantity it is of the same dimensions as the reciprocal of a pressure (or of a force per unit of area). The pressures concerned are usually measured in atmospheres (1 atmosphere = 1.014 × 10⁶ dynes per sq. cm.). For water the coefficient of compressibility, or the compression per atmosphere, is about 4.5 × 10^–5. This gives for $k$ the value 2.22 × 10¹⁰ dynes per sq. cm. The Young’s modulus and the rigidity of a fluid are always zero.

30. The relations between stress and strain in a material which is not isotropic are much more complicated. In such a material the Young’s modulus depends upon the direction of the tension, and its variations about a point are expressed by means of a surface of the fourth degree. The Poisson’s ratio depends upon the direction of the contracted lateral filaments as well as upon that of the longitudinal extended ones. The rigidity depends upon both the directions involved in the specification of the shearing stress. In general there is no simple relation between the Young’s moduluses and Poisson’s ratios and rigidities for assigned directions and the modulus of compression. Many materials in common use, all fibrous woods for example, are actually aeolotropic (that is to say, are not isotropic), but the materials which are aeolotropic in the most regular fashion are natural crystals. The elastic behaviour of crystals has been studied exhaustively by many physicists, and in particular by W. Voigt. The strain-energy-function is a homogeneous quadratic function of the six strain-components, and this function may have as many as 21 independent coefficients, taking the place in the general case of the 2 coefficients $\lambda ,\mu$ which occur when the material is isotropic—a result first obtained by George Green in 1837. The best experimental determinations of the coefficients have been made indirectly by Voigt by means of results obtained in the theories of the torsion and flexure of aeolotropic bars.

31. Limits of Elasticity.—A solid body which has been strained by considerable forces does not in general recover its original size and shape completely after the forces cease to act. The strain that is left is called set. If set occurs the elasticity is said to be “imperfect,” and the greatest strain (or the greatest load) of any specified type, for which no set occurs, defines the “limit of perfect elasticity” corresponding to the specified type of strain, or of stress. All fluids and many solid bodies, such as glasses and crystals, as well as some metals (copper, lead, silver) appear to be perfectly elastic as regards change of volume within wide limits; but malleable metals and alloys can have their densities permanently increased by considerable pressures. The limits of perfect elasticity as regards change of shape, on the other hand, are very low, if they exist at all, for glasses and other hard, brittle solids; but a class of metals including copper, brass, steel, platinum are very perfectly elastic as regards distortion, provided that the distortion is not too great. The question can be tested by observation of the torsional elasticity of thin fibres or wires. The limits of perfect elasticity are somewhat ill-defined, because an experiment cannot warrant us in asserting that there is no set, but only that, if there is any set, it is too small to be observed.

32. A different meaning may be, and often is, attached to the phrase “limits of elasticity” in consequence of the following experimental result:—Let a bar be held stretched under a moderate tension, and let the extension be measured; let the tension be slightly increased and the extension again measured; let this process be continued, the tension being increased by equal increments. It is found that when the tension is not too great the extension increases by equal increments (as nearly as experiment can decide), but that, as the tension increases, a stage is reached in which the extension increases faster than it would do if it continued to be proportional to the tension. The beginning of this stage is tolerably well marked. Some time before this stage is reached the limit of perfect elasticity is passed; that is to say, if the load is removed it is found that there is some permanent set. The limiting tension beyond which the above law of proportionality fails is often called the “limit of linear elasticity.” It is higher than the limit of perfect elasticity. For steel bars of various qualities J. Bauschinger found for this limit values varying from 10 to 17 tons per square inch. The result indicates that, when forces which produce any kind of strain are applied to a solid body and are gradually increased, the strain at any instant increases proportionally to the forces up to a stage beyond that at which, if the forces were removed, the body would completely recover its original size and shape, but that the increase of strain ceases to be proportional to the increase of load when the load surpasses a certain limit. There would thus be, for any type of strain, a limit of linear elasticity, which exceeds the limit of perfect elasticity.

33. A body which has been strained beyond the limit of linear elasticity is often said to have suffered an “over-strain.” When the load is removed, the set which can be observed is not entirely permanent; but it gradually diminishes with lapse of time. This phenomenon is named “elastic after-working.” If, on the other hand, the load is maintained constant, the strain is gradually increased. This effect indicates a gradual flowing of solid bodies under great stress; and a similar effect was observed in the experiments of H. Tresca on the punching and crushing of metals. It appears that all solid bodies under sufficiently great loads become “plastic,” that is to say, they take a set which gradually increases with the lapse of time. No plasticity is observed when the limit of linear elasticity is not exceeded.

34. The values of the elastic limits are affected by overstrain. If the load is maintained for some time, and then removed, the limit of linear elasticity is found to be higher than before. If the load is not maintained, but is removed and then reapplied, the limit is found to be lower than before. During a period of rest a test piece recovers its elasticity after overstrain.

35. The effects of repeated loading have been studied by A. Wöhler, J. Bauschinger, O. Reynolds and others. It has been found that, after many repetitions of rather rapidly alternating stress, pieces are fractured by loads which they have many times withstood. It is not certain whether the fracture