20 HISTORICAL INTRODUCTION. ment of scientific thought, and, having served its turn as a means of introducing generality into the subject, it must give place again to a still more general method. Saint-Venant as the champion of rari-constancy has repeatedly urged an objection against Green's method which its supporters do not appear to have met directly. The step objected to is the supposed possibility of expanding the energy-function in terms of the strain-components, and the retention of the second term. It is true that, as a proposition in pure mathematics, the step is unjustifiable. We have no right to assume that because one quantity depends upon another, and the first vanishes, and has a minimum value when the second vanishes, that, therefore, the first can be expanded in powers of the second, and terms of the second order occur. Many examples could be given to the contrary. But it is different in the case of elasticity. There is a definite physical reason, not stated by Green, and not generally stated in that connexion by his followers, viz. :--that experiment shews that the stress, in an elastic solid strained at constant temperature, or executing small vibrations, is a linear function of the strain, and it follows from this, analytically, that the potential energy of strain if a function of the strains at all, is a quadratic function of the strains, when the latter are small. That the potential energy is a function of the strains in these two cases is a proposition in Thermodynamics, first proved by Sir W. Thomson. We have just seen that the modern theory of elasticity rests upon the generalised Hooke's Law, as a fundamental datum given in experience. It is therefore necessary to pay some attention to the history of science in respect of this law. Its discovery by Hooke and Marriotte has already been noticed, but the experiments which led them to it were not of a very conclusive character. James Bernoulli, the discoverer of the elastic line, challenged it in 1744. The mathematicians of the 18th century assumed the linearity of the relation between tension and extension, whenever they needed it. For this case, Young gave precision to the law by the introduction of his modulus. Hodgkinson's experiments on cast-iron led him to conclude that, for this material at any rate, the law does not hold good. The discoverers of the general equations of elasticity, Navier, Poisson, and Cauchy, could all have deduced it from their molecular hypothesis if they had paid attention to the point, but they did not. This was reserved for Saint-Venant and Lamé. The point was really settled in 1845, when Sir G. Stokes remarked that the capacity of all solids to execute isochronous vibrations proves that the stress-strain relations must be linear for the very small displacements involved. It is sufficient for the mathematical theory as at present developed to know that the law is true for infinitesimal strains. It is a matter of interest, for possible future developments, to know further that, for all solids, (except cast-iron and perhaps some other cast metals), the law represents the stress-strain relation, as accurately as experiment can tell, for finite strains within the elastic limits. Now just as the generalised Hooke's Law was introduced into the mathematical theory from the analytical rather than the physical side, so almost the whole machinery of coefficients of elasticity, expressing the law, comes from the same source. Young's modulus, as a coefficient, is practically in the old theories of beams, in vogue before the time of Young. The rigidity, or coefficient of resistance to shearing strain, was in mathematical memoirs, (of course without a name), before it was suggested by Vicat1 and defined by Navier. The whole set of 21 coefficients of Green's energy-function remained unnamed till the appearance of Rankine's paper of 18553. But, after the introduction of A's and B's to express properties of matter, the physicist has come forward with an explanation as to what property of matter is expressed by A or B, his work has been a nomenclature of the A's and B's depending on something concrete which they really express, or the discovery of relations between the coefficients and some possible new set expressing simpler properties. In the theory of isotropic solids there occur two constants at most, say the K and k of Cauchy's first memoir. If Poisson's ratio be 1, k=2K. Cauchy's equations involving these constants are obtained by means of rather arbitrary assumptions. Different writers use different constants, which can be expressed in terms of Cauchy's. Navier and Poisson use a single constant, and so in other writings 1 Recherches expérimentales sur...la rupture'. Annales des ponts et chaussées, Mémoires 1833. 2 In the second edition of his Leçons, 1833. 3 On Axes of Elasticity and Crystalline Forms'. Rankine's Miscellaneous Scientific Papers. 22 HISTORICAL INTRODUCTION. does Cauchy. Lamé and Clapeyron use a single constant. Meanwhile Young's modulus is already defined physically. Presently comes Vicat with a physical definition of the rigidity. What is the relation of these physical constants to the coefficients in the elastic equations? There is no answer, but Green appears instead with two new constants A and B which he shews depend on the velocities of plane waves in the solid. Sir G. Stokes follows with again two new constants, defined, this time, from physical considerations. One is Vicat's rigidity, the other is the modulus of compression, or the ratio of a hydrostatic pressure applied uniformly to a solid to the cubical compression it produces. Then comes Lamé with his constants μ and λ, obtained rather in the manner of Cauchy's K and k, easily expressible in terms of those of Sir G. Stokes or Green, of whose writings he appears ignorant, is in fact the rigidity. Kirchhoff follows with his K and 0, of which K is the rigidity and a number, these are introduced like Green's A and B as coefficients in the energy-function. In reading any memoir it is necessary to have some acquaintance with six constants, the more or less arbitrary pair used by the writer of the memoir, the modulus of compression, the rigidity, Young's modulus, and Poisson's ratio. μ For æolotropic solids the matter is much simplified by the comparative smallness of the literature. Green introduced his 21 coefficients, and gave little explanation of them. Franz Neumann1 was the first to use the coefficients of Green's energy-function to He assumed that express the elastic properties of crystals. crystallographic symmetry corresponds to symmetry in elastic quality, and he thence shewed how to find the proper reductions in the number of the constants for the holohedral forms of the six classes of crystals, and, for systems having three planes of symmetry, he further shewed how to express the Young's modulus of the material, in a given direction, in terms of the coefficients. This theory has received much attention at the hands of Saint-Venant. Prof. Voigts has extended Neumann's work so as to include the principal hemihedral crystalline forms, 1 Vorlesungen über die Theorie der Elasticität der festen Körper und des Lichtäthers, 1885. The lectures were delivered in 1857-8. 2 'Mémoire sur la distribution des élasticités autour de chaque point'. Liouville's Journal de Mathématiques, vIII. 1863. 3 Wiedemann's Annalen, xvi. 1882. and has developed the theories of flexure and torsion, so as to obtain experimental methods for determining the constants of crystals with high degrees of symmetry. We have already seen how his experiments throw light on the constant controversy. The most important of Saint-Venant's researches, in this part of the subject, relates to the formula, which gives Young's modulus for any direction in an æolotropic solid with three planes of symmetry. Neumann had shewn that the modulus in any direction is proportional to the inverse fourth power of the radius-vector of a certain quartic surface, the coefficients in which are functions of the coefficients of elasticity1. SaintVenant proved that this radius-vector has 13 maxima and minima, but, if certain inequalities among the elastic coefficients be fulfilled, all but three are imaginary. It appears not unlikely that the maxima and minima of the Young's modulus should belong to principal axes of symmetry only. Saint-Venant also investigated the values of Poisson's ratio for extension in the direction of one axis, and contraction in that of another. He applied these researches to obtain formulæ that might prove useful in the case of timber and laminated metals, which have a certain æolotropic character without being crystalline. Another matter, to which he drew attention2, was the possibility of the directions of the principal axes of symmetry of contexture of a material, varying, from point to point, according to a definite law, so that, when suitable curvilinear coordinates are employed the stresses may be expressed in terms of the strains by formula which hold for all points, and he applied this theory to obtain results suitable for the explanation of certain piezometer experiments by Regnault, in which a shell of metal, forming part of the apparatus, probably has such a kind of solotropy. Two other points should be noticed in connexion with the elastic constants. One is that they vary with the temperature. In general a rise of temperature is accompanied by a decrease in the values of the constants. This point has been established chiefly by the experiments of Wertheim3, Kohlrausch and Mr 1 See Saint-Venant's 'Annotated Clebsch'. Note du § 16. 2. Sur les divers genres d'homogénéité des corps solides'. Liouville's Journal, 1865. 3 Recherches sur l'élasticité'. Annales de chimie, XII. 1844. 4 Pogg. Ann. CXLI. 1870. 24 HISTORICAL INTRODUCTION. The other is that the constants in the Donald McFarlane1. equations of vibration are not identical with those in the equations of equilibrium. This may be illustrated by a reference to Laplace's In the celebrated correction of the Newtonian velocity of sound. case of vibrations, the changes of state follow the adiabatic law, no heat being gained or lost by any element; in the case of strain gradually produced at constant temperature, the changes of state, following the isothermal law, differ from those that have place in a vibrating solid. The moduluses in the two cases are called by Sir W. Thomson kinetic and static moduluses respectively, and the latter are a little smaller than the former, but the ratio is very much nearer to unity for solids than for air. This point seems to have been first investigated by Lagerhjelm in 1827. Before passing to the consideration of problems, it is proper to notice some other matters connected with the general theory. These are the thermo-elastic equations of Neumann and Duhamel, the transformation of the equations of elasticity to orthogonal curvilinear coordinates, the theory of the propagation of disturbances by wave-motion in an unlimited elastic solid medium, and the general theory of the free vibrations of solids. One method by which the ordinary equations of elasticity have been obtained is, as we have seen, to assume that an elastic solid behaves like a system of material points, between which are forces of attraction or repulsion, and to estimate the stress thence arising when alterations are made in the intermolecular distances. When the temperature is variable, the force cannot be taken simply a function of the distance. Duhamel3 assumed that there is in this case an additional term in the force, proportional to the increase of temperature, and he thence obtained equations for the equilibrium of a solid strained by unequal heating. Franz Neumann* about the same time obtained similar equations by a method, which amounted to assuming that in a small part of a solid, so strained, there is a uniform elastic pressure proportional to the 1 Quoted by Sir W. Thomson, art. Elasticity, Encyc. Brit. and Math. and Phys. Papers, vol. 111. 2 See Todhunter and Pearson, vol. 1. art. 370. 3 Mémoire sur le calcul des actions moléculaires développées par les change- |