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.
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.
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.
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.
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.
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...
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.
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.
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.
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.
From Hooke's Law and from considerations of symmetry [Stokes] concluded that pressure equal in all directions round a point is attended by a proportional compression without shear, and that shearing stress is attended by a corresponding proportional shearing strain.
As an experimental basis for Hooke's Law [Stokes] cited the fact that bodies admit of being thrown into states of isochronous vibration.
The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Philosophy than in material progress, in trying to understand the world than in trying to make it more comfortable.
[D]iscussions... concerning the number and meaning of the elastic constants have thrown light on most recondite questions concerning the nature of molecules and the mode of their interaction.
The... fact that most great advances in Natural Philosophy have been made by men who had a first-hand acquaintance with practical needs and experimental methods has often been emphasized; and, although the names of Green, Poisson, Cauchy show that the rule is not without important exceptions, yet it is exemplified well in the history of our science.
Whenever, owing to any cause, changes take place in the relative positions of the parts of a body the body is said to be "strained." A very simple example of a strained body is a stretched bar.
Let is a number (generally a very small fraction) which is called the extension...