A Treatise on the Mathematical Theory of Elasticity

"The present treatise is the outcome of a suggestion made to me some years ago by Mr R. R. Webb that I should assist him in the preparation of a work on Elasticity. He has unfortunately found himself unable to proceed... and I have therefore been obliged to take upon myself the whole of the... responsibility. I wish to acknowledge... the debt that I owe to him as a teacher of the subject, as well as... for many valuable suggestions..."

"The division of the subject adopted is that... by Clebsch in his classical treatise, where a clear distinction is drawn between exact solutions for bodies all whose dimensions are finite and approximate solutions for bodies some of whose dimensions can be regarded as infinitesimal. The present volume contains the general mathematical theory of the elastic properties of the first class of bodies, and I propose to treat the second class in another volume."

"At Mr Webb's suggestion, the exposition of the theory is preceded by an historical sketch of its origin and development. Anything like an exhaustive history has been rendered unnecessary by the work of the late Dr Todhunter as edited by Prof. Karl Pearson, but it is hoped that the brief account given will at once facilitate the comprehension of the theory and add to its interest."

"In the analysis of strain I have thought it best to follow Thomson and Tait's Natural Philosophy, beginning with the geometrical or rather algebraical theory of finite homogeneous strain, and passing to the physically most important case of infinitesimal strain."

"The discussion of the stress-strain relations rests upon as an axiom generally verified in experience, and on Sir W. Thomson['s] thermodynamical investigation of the existence of the energy-function."

"The theory of elastic crystals adopted is that which has been elaborated by the researches of F. E. Neumann and W. Voigt."

"The conditions of rupture or rather of safety of materials are as yet so little under stood that it seemed best to give a statement of the various theories that have been advanced without definitely adopting any of them."

"In most of the problems considered in the text Saint-Venant's "greatest strain" theory has been provisionally adopted. In connexion with this theory I have endeavoured to give precision to the term ""."

"Among general theorems I have included an account of the deduction of the theory from Boscovich's point-atom hypothesis. This is rendered necessary partly by the controversy that has raged round the number of independent elastic constants, and partly by the fact that there exists no single investigation of the deduction in question which could now be accepted by mathematicians."

"[With regard to] Saint-Venant's theory of the equilibrium of beams... In spite of the work of Prof. Pearson it seems not yet to be understood by English mathematicians that the cross-sections of a bent beam do not remain plane. The old-fashioned notion of a bending moment proportional to the curvature resulting from the extensions and contractions of the fibres is still current. Against the venerable bending moment the modern theory has nothing to say, but it is quite time that it should be generally known that it is not the whole stress, and that the strain does not consist simply of extensions and contractions of the fibres. In explaining the theory I have followed Clebsch's mode of treatment, generalising it so as to cover some of the classes of aeolotropic bodies treated by Saint-Venant."

"[W] are occupied with the principal analytical problems presented by elastic theory. The theory leads in every special case to a system of partial differential equations, and the solution of these subject to conditions given at certain bounding surfaces is required. The general problem is that of solving the general equations with arbitrary conditions at any given boundaries. In discussing this problem I have made extensive use of the researches of Prof. Betti of Pisa, whose investigations are the most general that have yet been given..."

"The case of a solid bounded by an infinite plane and otherwise unlimited is investigated on the lines laid down by Signor Valentino Cerruti, whose analysis is founded on Prof. Betti's general method, and some of the most important particular cases are worked out synthetically by M. Boussinesq's method of potentials. In this connexion I have introduced the last-mentioned writer's theory of "local perturbations", a theory which gives the key to Saint-Venant's "principle of the elastic equivalence of statically equipollent systems of load"."

"The student without previous acquaintance with the subject is advised in all cases to provide the required proofs. It is hoped that he will not then fail to understand the subject for lack of examples, nor waste his time in mere problem grinding."

"The Mathematical Theory of Elasticity is occupied with an attempt to reduce to calculation the state of strain, or relative displacement, within a solid body which is subject to the action of an equilibrating system of forces, or is in a state of slight internal relative motion, and with endeavours to obtain results which shall be practically important in applications to architecture, engineering, and all other useful arts in which the material of construction is solid."

"Alike in the experimental knowledge obtained, and in the analytical methods and results, nothing that has once been discovered ever loses its value or has to be discarded; but the physical principles come to be reduced to fewer and more general ones, so that the theory is brought more into accord with that of other branches of physics, the same general dynamical principles being ultimately requisite and sufficient to serve as a basis for them all."

"[A]lthough, in the case of Elasticity, we find frequent retrogressions on the part of the experimentalist, and errors on the part of the mathematician, chiefly in adopting hypotheses not clearly established or... discredited, in pushing to extremes methods merely approximate, in hasty generalizations, and in misunderstandings of physical principles, yet we observe a continuous progress in all the respects mentioned when we survey the history of the science from the initial enquiries of Galileo to the conclusive investigations of Saint-Venant and Lord Kelvin."

"The first mathematician to consider the nature of the resistance of solids to rupture was Galileo. Although he treated solids as inelastic, not being in possession of any law connecting the displacements produced with the forces producing them, or of any physical hypothesis capable of yielding such a law, yet his enquiries gave the direction which was subsequently followed by many investigators."

"[Galileo] endeavoured to determine the resistance of a beam, one end of which is built into a wall, when the tendency to break it arises from its own or an applied weight; and he concluded that the beam tends to turn about an axis perpendicular to its length, and in the plane of the wall. This problem, and, in particular, the determination of this axis is known as Galileo's problem."

"[T]he two great landmarks are the discovery of in 1660, and the formulation of the general equations by Navier in 1821."

"provided the necessary experimental foundation for the theory. When the general equations had been obtained, all questions of the small strain of elastic bodies were reduced to a matter of mathematical calculation."

"Hooke and Mariotte occupied themselves with the experimental discovery of what we now term stress-strain relations. Hooke gave in 1678 the famous law of proportionality of stress and strain which bears his name, in the words "Ut tensio sic vis; that is, the Power of any spring is in the same proportion with the Tension thereof." By "spring" Hooke means... any "springy body," and by "tension" what we should now call "extension," or, more generally, "strain." This law he discovered in 1660, but did not publish until 1676, and then only under the form of an anagram, ceiiinosssttuu. This law forms the basis of the mathematical theory of Elasticity."

"Hooke does not appear to have made any application of [his law] to the consideration of Galileo's problem. This application was made by Mariotte, who in 1680 enunciated the same law independently. He remarked that the resistance of a beam to flexure arises from the extension and contraction of its parts, some of its longitudinal filaments being extended, and others contracted. He assumed that half are extended, and half contracted. His theory led him to assign the position of the axis, required in the solution of Galileo's problem, at one-half the height of the section above the base."

"In the interval between the discovery of Hooke's law and that of the general differential equations of Elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibrations of bars and plates, and the stability of columns."

"The first investigation of any importance is that of the elastic line or elastica by James Bernoulli in 1705, in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practically involves the result that the resistance to bending is a couple proportional to the of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods."

"As soon as the notion of a flexural couple proportional to the curvature was established it could be noted that the work done in bending a rod is proportional to the square of the curvature."

"Daniel Bernoulli suggested to Euler that the differential equation of the elastica could be found by making the integral of the square of the curvature taken along the rod a minimum... Euler, acting on this... was able to obtain the differential equation of the curve..."

"Euler pointed out... that the rod, if of sufficient length and vertical when unstrained, may be bent by a weight attached to its upper end... [and was led] to assign the least length of a column in order that it may bend under its own or an applied weight. Lagrange followed and used his theory to determine the strongest form of column. These two... found [the] length which a column must attain to be bent by its own or an applied weight, and... that for shorter lengths it will be simply compressed, while for greater lengths it will be bent. These researches are the earliest in... elastic stability."

"In Euler's work on the elastica the rod is thought of as a line of particles which resists bending. The theory of the flexure of beams of finite section was considered by Coulomb... [by investigating] the equation of equilibrium obtained by resolving horizontally the forces which act upon the part of the beam cut off by one of its normal sections, as well as of the equation of moments. He... thus... obtain[ed] the true position of the "neutral line," or axis of equilibrium, and he also made a correct calculation of the moment of the elastic forces. His theory of beams is the most exact of those [that assume] the stress in a bent beam arises wholly from the extension and contraction of its longitudinal filaments, and... Hooke's Law."

"Coulomb was also the first to consider the resistance [although considered as nonelastic] of thin fibres to torsion... to which Saint-Venant refers under the name I'ancienne thiorie... Coulomb was [also] first to [consider] strain we now call shear, though he considered it in connexion with rupture only... when the shear [permanent set, not an elastic strain] of the material is greater than a certain limit."

"Except Coulomb's, the most important work of the period for the general mathematical theory is the physical discussion of elasticity by Thomas Young. ...[Young,] besides defining his modulus of elasticity, was the first to consider shear as an elastic strain. He called it "detrusion," and noticed that the elastic resistance of a body to shear, [as opposed to] its resistance to extension or contraction, are in general different; but he did not introduce a distinct modulus of rigidity to express resistance to shear. He defined "the modulus of elasticity of a substance" as "a column of the same 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."

"During the first period in the history of our science (1638—1820) while these various investigations of special problems were being made, there was a cause at work which was to lead to wide generalizations. This cause was physical speculation concerning the constitution of bodies. In the eighteenth century the Newtonian conception of material bodies, as made up of small parts which act upon each other by means of central forces, displaced the Cartesian conception of a plenum pervaded by "vortices." Newton regarded his "molecules" as possessed of finite sizes and definite shapes, but his successors gradually simplified them into material points. The most definite speculation of this kind is that of Boscovich, for whom the material points were nothing but persistent centres of force. To this order of ideas belong Laplace's theory of capillarity and Poisson's first investigation of the equilibrium of an "elastic surface," but for a long time no attempt seems to have been made to obtain general equations of motion and equilibrium of elastic solid bodies."

"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 plates, 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 distinction between shear and extension was a preliminary to a general theory of strain; the recognition of forces across the elements of 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 bare and plates, was a foreshadowing of the employment of differential equations of displacement; the Newtonian conception of the constitution of bodies, combined with Hooke's Law, offered means for the formation of such equations; and the generalization of the principle of 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 the clear aim to discover the laws by which phenomena are connected with each other, 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."

"In... 1821... Fresnel announced his conclusion that the observed facts in regard to the interference of polarised light could be explained only by the hypothesis of transverse vibrations. He showed how a medium consisting of "molecules " connected by central forces might be expected to execute such vibrations and to transmit waves of the required type. Before the time of Young and Fresnel such examples of transverse waves as were known—waves on water, transverse vibrations of strings, bars, membranes and plates—were in no case examples of waves transmitted through a medium; and neither the supporters nor the opponents of the undulatory theory of light appear to have conceived of light waves otherwise than as "longitudinal " waves of condensation and rarefaction, of the type rendered familiar by the transmission of sound."

"The theory of elasticity, and, in particular, the problem of the transmission of waves through an elastic medium now attracted the attention of... Cauchy and Poisson—the former a discriminating supporter, the latter a sceptical critic of Fresnel's ideas. In the future the developments of the theory of elasticity were to be closely associated with the question of the propagation of light, and these developments arose in great part from the labours of these two savants."

"By the Autumn of 1822 Cauchy had discovered most of the elements of the pure theory of elasticity. ...[H]e had generalized the notion of hydrostatic pressure, and he had shown that the stress is expressible by means of six component stresses, and also by means of three purely normal tractions across a certain triad of planes which cut each other at right angles—the "principal planes of stress.""

"[Cauchy] had shown also how the differential coefficients of the three components of displacement can be used to estimate the extension of every linear element of the material, and had expressed the state of strain near a point in terms of six components of strain, and also in terms of the extensions of a certain triad of lines which are at right angles to each other—the "principal axes of strain.""

"[Cauchy] had determined the equations of motion (or equilibrium) by which the stress-components we connected with the forces that are distributed through the volume and with the kinetic reactions. By means of relations between stress-components and strain-components, he had eliminated the stress-components from the equations of motion and equilibrium, and had arrived at equations in terms of the displacements."

"Cauchy obtained his stress-strain relations for isotropic materials by means of two assumptions, viz. : (1) that the relations in question are linear, (2) that the principal planes of stress are normal to the principal axes of strain."

"[Cauchy's] equations... are those which are now admitted for isotropic solid bodies. The methods used in these investigations are quite different from... Navier's... no use is made of the hypothesis of material points and central forces. ...Navier's equations contain a single constant to express the elastic behaviour of a body, while Cauchy's contain two such constants."

"At a later date Cauchy extended his theory to the case of crystalline bodies, and he then made use of the hypothesis of material points between which there are forces of attraction or repulsion."

"Clausius criticized the restrictive conditions which Cauchy imposed upon the arrangement of his material points, but he argued that these conditions are not necessary for the deduction of Cauchy's equations."

"Poisson's] April, 1828... memoir is very remarkable... like Cauchy, [he] first obtains the equations of equilibrium in terms of stress-components, and then estimates the traction across any plane resulting from the "intermolecular" forces. The expressions... involve summations with respect to all the "molecules," situated within the region of "molecular" activity of a given one. Poisson... assumes... summations with respect to angular space about the given "molecule," but not... with respect to distance... The equations of equilibrium and motion of isotropic elastic solids... thus obtained are identical with Navier's."

"Poisson assumed that the irregular action of the nearer molecules may be neglected, in comparison with the action of the remoter ones, which is regular. This assumption is the text upon which Stokes afterwards founded his criticism of Poisson. As we have seen, Cauchy arrived at Poisson's results by the aid of a different assumption. Clausius held that both Poisson's and Cauchy's methods could be presented in unexceptionable forms."

"The revolution which Green effected in the elements of the theory is comparable in importance with that produced by Navier's discovery of the general equations. Starting from what is now called the Principle of the Conservation of Energy he propounded a new method of obtaining these equations."

"[Green] stated his principle and method in the following words:— "In whatever way the elements of any material system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function. But this function being known, we can immediately apply the general method given in the Mécanique Analytique, and which appears to be more especially applicable to problems that relate to the motions of systems composed of an immense number of particles mutually acting upon each other. One of the advantages of this method, of great importance, is that we are necessarily led by the mere process of the calculation, and with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied.""

"The function here spoken of, with its sign changed, is the potential energy of the strained elastic body per unit of volume, expressed in terms of the components of strain; and the differential coefficients of the function, with respect to the components of strain, are the components of stress. Green supposed the function to be capable of being expanded in powers and products of the components of strain. He therefore arranged it as a sum of homogeneous functions of these quantities of the first, second and higher degrees. Of these terms, the first must be absent, as the potential energy must be a true minimum when the body is unstrained; and, as the strains are all small, the second term alone will be of importance. From this principle Green deduced the equations of Elasticity, containing in the general case 21 constants. In the case of isotropy there are two constants, and the equations are the same as those of Cauchy's first memoir."

"Lord Kelvin has based the argument for the existence of Green's strain-energy-function on the First and Second Laws of Thermodynamics. From these laws he deduced the result that, when a solid body is strained without alteration of temperature, the components of stress are the differential coefficients of a function of the components of strain with respect to these components severally. The same result can be proved to hold when the strain is effected so quickly that no heat is gained or lost by any part of the body."

"Poisson's theory leads to the conclusions that the resistance of a body to compression by pressure uniform all round it is two-thirds of the of the material, and that the resistance to shearing is two-fifths of the Young's modulus. He noted a result equivalent to the first of these, and the second is virtually contained in his theory of the torsional vibrations of a bar."

"The observation that resistance to compression and resistance to shearing are the two fundamental kinds of elastic resistance in isotropic bodies was made by Stokes, and he introduced definitely the two principal moduluses of elasticity... the "modulus of compression" and the "rigidity," as they are now called."