noticed that the elastic resistance of a body to shear, and its resistance to extension or compression, are in general different; but he did not expressly introduce a new modulus of rigidity for this resistance. He defined "the modulus of elasticity of a substance1" as a column of the substance capable of producing a pressure on its base, which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length. What we now call "Young's modulus" is the weight of this column per unit of area of its base. This introduction of a definite physical concept, associated with the coefficient of elasticity, which descends as it were from a clear sky on the reader of mathematical memoirs, marks an epoch in the history of the science. In the literature of this, the first period in the history of our subject, there are many discussions of the physical cause of elasticity, the philosophers, generally, either following Descartes, and believing in space continuously filled and a subtle æther that is in the pores of bodies, or else following the suggestion of Newton, that all the interactions between parts of bodies can be reduced to attracting and repelling forces between the ultimate molecules, which operate immediately, without any intervening mechanism. But no attempt appears to have been made to deduce general equations of motion and equilibrium from either of these hypotheses. At the end of the year 1820, the fruit of all the ingenuity expended on elastic problems might be summed up as—an inadequate theory of flexure, an erroneous theory of torsion, an unproved theory of the vibrations of bars, and the definition of Young's modulus. But such an estimate would give a very wrong impression of the value of the older researches. The recognition of the fact, that there is a fundamental difference between shear and extension, was a preliminary to a general theory of strain; the discovery of forces across a section of a beam, producing a resultant, was a step towards a theory of stress; the use of differential equations for the deflexion of a bent beam and the vibrations of bars and plates, was a foreshadowing of the employment of differential equations of displacement; the suggestion of Newton and the enunciation of Hooke's law, offered means for the formation 1 Loc. cit. This was given in section Ix. of vol. II. of the first edition, and omitted in Kelland's edition, but it is reproduced in the Miscellaneous Works of Dr Young. 6 HISTORICAL INTRODUCTION. of such equations; and the generalisation of the principle of virtual work in the Mécanique Analytique threw open a broad path to discovery in this as in every other branch of mathematical physics. Physical Science had emerged from its incipient stages with definite methods, of hypothesis and induction and of observation and deduction, with a clear aim, to explain facts, and with a fund of analytical processes of investigation. This was the hour for the production of general theories, and the men were not wanting. There are two subjects, usually included in the general theory of elasticity, which have an extended application to other branches of mathematical Physics, these are the analysis of strain and the The first gives general considerations as to analysis of stress. the kinematical expression of the possible deformations of the parts of any medium which can be treated as continuous, the second gives similar considerations relative to the kind of internal The foundation of both forces that can exist in such media. theories was laid by Cauchy in 1827, but he appears to have been in possession of some of the results as early as 1822, when he communicated an account of his researches to the Paris Academy1. Among his discoveries must be reckoned the determination of the stress at any point in terms of six component stresses, and of the strain, whether finite or infinitesimal, in terms of six component strains, the properties of the stressquadric, stress-ellipsoid, strain-quadric, and elongation-quadric, and the existence of principal stresses and principal extensions. Results equivalent to some of Cauchy's were discovered independently by Lamé, who developed somewhat the geometrical study of distributions of stress by means of the properties of certain quadric surfaces. Cauchy's expressions of the six com 1 Bulletin...Philomatique, 1823. 2 2 See Exercices de Mathématiques, 1827, in which are the following memoirs: 'De la pression ou tension dans un corps solide', 'Sur la condensation et la dilatation des corps solides', and Exercices de Mathématiques 1828, in which is a memoir 'Sur quelques theorèmes relatifs à la condensation ou à la dilatation des corps'. 3 The assumption involved in this reduction does not appear to have been noticed by writers on elastic theory. The fact that a medium is possible in which it does not hold good appears to have been first noticed in connexion with Electrodynamics. 4 Lamé and Clapeyron, Mémoire sur l'équilibre intérieur des corps solides homogènes'. Mém....par divers savans, iv. 1833. The date of the memoir is at least as early as 1828. ponents of finite strain are practically those of Green', and Saint-Venant, but the latter was the first to consider them minutely. To Saint-Venant more than anyone else belongs the credit of the adequate discussion of shear; he was the first mathematician to call attention to its importance as a specific kind of strain; previously to his time the quantities we should now call shears made their appearance simply as mathematical expressions. Sir W. Thomson further simplified the discussion of strain by the introduction of his strain-ellipsoid, and the kinematical theory reaches its highest development in Thomson and Tait's Natural Philosophy, Part I. To a modern reader it might appear that the analysis of stress and strain is a necessary preliminary to a general theory of elasticity, but historically this was not the order in which discoveries were made. The investigation of the general equations by Navier does not depend on any such analysis; Poisson's investigation involves an analysis of stress, but not of strain, Green's an analysis of strain, but not of stress. There are in fact three fundamental methods of arriving at these equations. The first consists in assuming a law as to the character of intermolecular force, and deducing the differential equations of displacement from the equations of equilibrium of a single displaced "molecule". This is Navier's method. The second method consists in forming differential equations of equilibrium of any element in terms of the stresses exerted upon it by the surrounding matter, and then, by means of relations between stress and relative displacement, eliminating the stress-components from these equations. The required relations may be assumed, as in Cauchy's first investigation, or deduced from experiment, as by Sir G. Stokes, or calculated from an assumed law of intermolecular force, as by Poisson and Cauchy. The third method consists in writing down an expression for the energy of the strained solid, and deducing 1 'On the Laws of Reflexion and Refraction of Light at the common surface of two non-crystallized media', Camb. Phil. Soc. Trans. vII. 1837. See also Math. Papers of the late George Green, 1871. 2 'Mémoire sur l'équilibre des corps solides', Comptes rendus, xxiv. 1847. The expressions referred to were given by Saint-Venant in 1844, see Todhunter and Pearson, vol. 1. art. 1614. 3 Leçons de Mécanique appliquée, 1837, 1838. See Todhunter and Pearson, vol. 1. arts. 1564, 1565, 1570. 4 Thomson and Tait, Nat. Phil. Part 1. arts. 155-190. 8 HISTORICAL INTRODUCTION. the equations by an application of the principle of Virtual Work. This method is due to Green, and has been followed by Kirchhoff and many English writers. Navier' was the first to investigate the general equations of equilibrium and vibrations of elastic solids. He set out from the hypothesis which we have ascribed to Newton, that the elastic reactions arise from variations in the intermolecular forces, consequent upon changes in the molecular configuration. He assumed that the force between two molecules, whose distance is slightly increased, is proportional to the product of the increase in the distance and some function of the initial distance. His method consists in forming an expression for the component in any direction of all the forces, that act upon a displaced "molecule", and thence the equations of motion of the molecule. The equations are thus obtained in terms of the displacements of the molecule. The solid is assumed to be isotropic, and the equations obtained contain a single elastic constant. Navier next formed an expression for the work done in a small relative displacement by all the forces which act upon a molecule; this he described as the sum of the moments in the sense of the Mécanique Analytique of the forces exerted by all the other molecules on a particular molecule. He deduced, by an application of the Calculus of Variations, not only the differential equations previously obtained, but also the boundary-conditions that hold at the surface of the body. This memoir is very important as the first general investigation of its kind, but its arguments would not now be admitted. In the first place the expression for the force between two molecules, after displacement, is incorrect; in the second place the expression for the component force in any direction, acting on a molecule, is wrongly discussed. This expression involves a triple summation, and Navier replaced the summations by integrations. It appears from subsequent investigations by Cauchy and Poisson that this step is unnecessary, and, if the force between two molecules be taken simply a function of their distance, it leads to absurd results when worked out correctly. Cauchy gave three ways of arriving at the equations, of which 1 Mém. Acad. Sciences, VII. Paris, 1827. The memoir was read in 1822. consists of a very large number of material points, with a law of force between pairs some function of their distance. In the first1 of these "molecular" memoirs an expression is formed for the forces that act upon a single 'molecule', and the differential equations deduced; in the case of isotropy these contain two constants. In the second expressions are formed for the stresses across any plane drawn in the solid. If the initial state be one of zero stress, and the solid isotropic, the stress will be expressed in terms of the strain by means of a single constant, and one of the constants of the preceding memoir must vanish. The equations are then identical with those of Navier, but they are obtained without replacing summations by integrations. In like manner, in the general case of æolotropy, Cauchy finds 21 independent constants, of which 6 vanish identically if the initial state be one of zero stress. These points were not fully explained by Cauchy. Clausius3, however, has shewn that this is the meaning of his work. Clausius criticises the considerations of symmetry in molecular arrangement, by which Cauchy reduced his 15 constants to one in the case of isotropy, but the reduction can be effected by other methods, and the equations must be regarded as proved if the "molecular" hypothesis be admitted. 4 The first memoir by Poisson relating to the same subject was read before the Paris Academy on April 14th, 1828. The memoir is very remarkable for its numerous applications of the general theory to special problems, but the treatment of the general equations is inferior to Cauchy's. Like Cauchy, Poisson first obtains the equations of equilibrium in terms of stresscomponents, and then estimates the stress across any plane resulting from the intermolecular forces. The expressions for the stresses in terms of the strains involve summations with respect to all the molecules, situated within the region of molecular activity of a given one. Poisson rightly decides against replacing the summations by integrations, but he assumes that this can be done 1 'Sur l'équilibre et le mouvement d'un système de points matériels'. Exercices de Mathématiques, 1828. 2 'De la pression ou tension dans un système de points matériels', same volume. 36 Ueber die Veränderungen, welche in den bisher gebräuchlichen Formeln für das Gleichgewicht und die Bewegung elastischer fester Körper durch neuere Beobachtungen nothwendig geworden sind'. Pogg. Ann. 76, 1849. 4 'Mémoire sur l'équilibre et le mouvement des corps élastiques'. Mém. Paris Acad. VIII. 1829. |