A Treatise on the Mathematical Theory of Elasticity

"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."

"By a method analogous to that of Cauchy's first memoir, but resting on the above-stated experimental basis, [Stokes] deduced the equations with two constants which had been given by Cauchy and Green. Having regard to the varying degrees in which different classes of bodies—liquids, soft solids, hard solids—resist compression and distortion, he refused to accept the conclusion from Poisson's theory that the modulus of compression has to the rigidity the ratio 5 : 3. He pointed out that, if the ratio of these moduluses could be regarded as infinite, the ratio of the velocities of "longitudinal " and " transverse " waves would also be infinite, and then, as Green had already shown, the application of the theory to optics would be facilitated."

"The hypothesis of material points and central forces does not now hold the field. ...Of much greater importance have been the development of the atomic theory in Chemistry and of statistical molecular theories in Physics, the growth of the doctrine of energy, the discovery of electric radiation. It is now recognized that a theory of atoms must be part of a theory of the æther, and that the confidence which was once felt in the hypothesis of central forces between material points was premature. To determine the laws of the elasticity of solid bodies without knowing the nature of the æthereal medium or the nature of the atoms, we can only invoke the known laws of energy as was done by Green and Lord Kelvin; and we may place the theory on a firm basis if we appeal to experiment to support the statement that, within a certain range of strain, the strain-energy-function is a quadratic function of the components of strain, instead of relying, as Green did, upon an expansion of the function in series."

"The problem of curved plates or shells was first attacked from the point of view of the general equations of Elasticity by H. Aron. He expressed the geometry of the middle-surface by means of two parameters after the manner of Gauss, and he adapted to the problem the method which Clebsch had used for plates. He arrived at an expression for the potential energy of the strained shell which is of the same form as that obtained by Kirchhoff for plates, but the quantities that define the curvature of the middle-surface were replaced by the differences of their values in the strained and unstrained states."

"E. Mathieu adapted to the problem [of curved plates or shells ] the method which Poisson had used for plates. He observed that the modes of vibration possible to a shell do not fall into classes characterized respectively by normal and tangential displacements, and he adopted equations of motion that could be deduced from Aron's formula for the by retaining the terms that depend on the stretching of the middle-surface only."

"Lord Rayleigh... concluded from physical reasoning that the middle-surface of a vibrating shell remains unstretched, and determined the character of the displacement of a point of the middle-surface in accordance with this condition. The direct application of the Kirchhoff-Gehring method led to a formula for the potential energy of the same form as Aron's and to equations of motion and boundary conditions which were difficult to reconcile with Lord Rayleigh's theory. Later investigations have shown that the extensional strain which was thus proved to be a necessary concomitant of the vibrations may be practically confined to a narrow region near the edge of the shell, but that, in this region, it may be so adjusted as to secure the satisfaction of the boundary conditions while the greater part of the shell vibrates according to Lord Rayleigh's type."

"Whenever very thin rods or plates are employed in constructions it becomes necessary to consider the possibility of , and thus there arises the general problem of elastic stability. [T]he first investigations... of this kind were made by Euler and Lagrange. ...In all [isolated problems] two modes of equilibrium with the same type of external forces are possible, and the ordinary proof of the determinacy of the solution of the equations of Elasticity is defective."

"A general theory of elastic stability has been proposed by G. H. Bryan. He arrived at the result that the theorem of determinacy cannot fail except in cases where large relative displacements can be accompanied by very small strains, as in thin rods and plates, and in cases where displacements differing but slightly from such as are possible in a rigid body can take place, as when a sphere is compressed within a circular ring of slightly smaller diameter. In all cases where two modes of equilibrium are possible the criterion for determining the mode that will be adopted is given by the condition that the energy must be a minimum."

"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."

"Even in the more technical problems, such as the transmission of force and the resistance of bars and plates, attention has been directed, for the most part, rather to theoretical than to practical aspects of the questions. To get insight into what goes on in impact, to bring the theory of the behaviour of thin bars and plates into accord with the general equations—these and such-like aims have been more attractive... than endeavours to devise means for effecting economies in engineering constructions or to ascertain the conditions in which structures become unsafe."

"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 l_0 be the length before stretching, and l the length when stretched. Then (l - l_0)/l_0is a number (generally a very small fraction) which is called the extension..."

"Let e denote the extension of the bar, so that its length is increased in the ratio 1 + e : 1 ...[V]olume is increased by stretching the bar, but not in the ratio 1 + e : 1. When the bar is stretched longitudinally it contracts laterally... If the linear lateral contraction is e^\prime, the sectional area is diminished in the ratio (1 - e^\prime)^2 : 1, and the volume in question is increased in the ratio (1 + e) (1 - e^\prime)^2 : 1. In... a bar under tension e^\prime is a certain multiple of e, say \sigma e... [with] \sigma... about \frac{1}{3} or \frac{1}{4} for very many materials. If e is very small and e^2 is neglected, the areal contraction is 2\sigma e, and the cubical dilatation is (1 - 2\sigma)e."

"[M]easure the coordinate z along the length of the [vertical] bar. Any particle of the bar which has the coordinates x, y, z when the weight is not attached will move after the attachment of the weight into a new position. Let the particle which was at the origin move through a distance z_0, then the particle which was at (x, y, z) moves to the point of which the coordinates arex(1 - \sigma e), \qquad y(1 - \sigma e), \qquad z_0 + (z - z_0)(1 + e)."

"Love was the first investigator to present a successful approximation shell theory based on classical elasticity. To simplify the strain-displacement relationships and, consequently, the constitutive relations, Love [in this Treatise] introduced the following assumptions, known as the first approximations and commonly termed the Kirchoff-Love hypotheses..."

"[Love's] early books on elasticity, theoretical mechanics, and calculus have been used by many generations of students, whilst the much enlarged edition of his Treatise on Elasticity is a monumental work."

"History of science"

"A History of the Theory of Elasticity and of the Strength of Materials"

"Engineering"