elasticity-physics

"Riccati gives a characteristic paragraph with regard to the English theorists:...Non ci ha fenomeno in Natura, ch' eglino non ascrivano alle favorite attrazioni, da cui derivano la durezza, la fluidità, ed altre proprietà de' composti, e spezialmente la forza elastica ... [There is no phenomenon in Nature, which is not ascribed to the favorite attractions, from which is derived hardness, fluidity, and other properties of the compounds, and especially the elastic force. ...]"

"Riccati... will not enter into these disputes [as to current hypotheses as to the nature of elasticity]... For [in] his own theory he will not call to his assistance the aether of Descartes or the attractions of Newton. ...[H]e ...seems ignorant of and quotes Gravesande [Physices elementa mathematica experimentis confirmata, 1720.] to shew that the relation of extension to force is quite unknown... curious as he elsewhere cites Hooke..."

"Riccati states la mia novella sentenza [my new sentence]... Every deformation is produced by forza viva and this force is proportional to the deformation produced. ...The forza viva spent in producing a deformation remains in the strained body in the form of forza morta; it is stored up in the compressed fibres. Riccati comes to this conclusion after asking whether the forza viva so applied could be destroyed? That... he denies, making use strangely enough of the argument from design, a metaphysical conception such as he has told us ought not to be introduced into physics!La Natura anderebbe successivamente languendo, e la materia diverrebbe col lungo girare de' secoli una massa pigra, ed informe fornita soltanto d' impenetrabilità, e d' inerzia, e spogliata passo passo di quella forza (conciossiachè in ogni tempo una notabil porzione se ne distrugge) la quale in quantità, ed in misura era stata dal sommo Facitore sin dall' origine delle cose ad essa addostata per ridurre il presente Universo ad un ben concertato Sistema. [Nature would then be languishing, and matter would become a lazy, unformed mass with the long passage of centuries, and only provided impenetrability, and inertia, and stripped step by step of that force (because at any time a notable portion destroys it) which in quantity, and to an extent had been from the supreme Authority since the origin of the things, subjected to, in order to reduce the present Universe to a well-organized System.]"

"This paragraph... unit[es] the old theologico-mathematical standpoint, with the first struggling towards the modern conception of the . It is this principle of energy which la mia novella sentenza endeavours so vaguely to express, namely that the mechanical work stored up in a state of strain, must be equivalent to the energy spent in producing that state."

"Riccati... tells us that the forza viva must be measured by the square of the velocity. The consideration of the impact of bodies is more suggestive; the forza viva existing before impact is converted at the moment into forza morta and this re-converted into forza viva partly in the motion of either body as a whole, and partly in the vibratory motion of their parts, which we perceive in the sound vibrations they give rise to in the air."

"The importance of Riccati's work lies not in his practical results, which are valueless, but in his statement of method, and his desire to replace by a dynamical theory semi-metaphysical hypotheses. ...[H]is writings remind us... of Bacon, who in like fashion failed to obtain valuable results, although he was capable of discovering a new method. Euler's return to the semi-metaphysical hypothesis... is a distinct retrogression on Riccati's attempt, which had to wait till George Green's day before it was again broached."

"Gravesande in his Physices Elementa Mathematica [Vol.1; Vol. 2], 1720, explains elasticity by Newtonian attractions and repulsions. The... chapter... entitled De legibus elasticitatis [The laws of elasticity].... is of opinion that within the limits of elasticity, the force required to produce any extension is a subject for experiment only. ...he considers elastic cords, laminae and spheres (supposed built up of laminae), and finds the deflection of the beam in Galilei's problem proportional to the weight. He makes... no attempt to discuss the elastic curve."

"The direct impulse to investigate elastic problems... came to Euler from the Bernoullis."

"Galilei's problem had determined the direction of later researches... while James Bernoulli solved the problem of the elastic curve his nephew Daniel first obtained a differential equation which really does present itself in the consideration of the transverse vibrations of a bar."

"[In an Oct. 20, 1742 letter, Daniel Bernoulli] suggests for Euler's consideration the case of a beam with clamped ends, but states that the only manner in which he has himself found a solution of this "idea generalissima elasticarum" is "per methodum isoperimetricorum." He assumes the "vis viva potentialis laminae elasticae insita" must be a minimum, and thus obtains a differential equation of the fourth order, which he has not solved, and so cannot yet shew that this "aequatio ordinaria elasticae" is general.Ew. reflectiren ein wenig darauf ob man nicht konne sine interventu vectis die curvaturam immediate ex principiis mechanicis deduciren. Sonsten exprimire ich die vim vivam potentialem laminae elasticae naturaliter rectae et incurvatae durch \int ds/R^2, sumendo elementum ds pro constante et indicando radium osculi per R. Da Niemand die methodum isoperimetricorum so weit perfectionniret als Sie, werden Sic dieses problema, quo requiritur ut \int ds/R^2 faciat minimum, gar leicht solviren. [Ew. reflect a little on whether one can not deduce the curvature of the bar directly from the principles of mechanics. In the first place I express the actual elastic laminar potential, naturally right and yet curving, by \int ds/R^2, summing the element ds per constant radius of curvature R. Since no one has perfected the isoperimetric method as much as You, So this problem, which requires that \int ds/R^2 be minimum, might be easily solved.]"

"Bernoulli writes... to Euler... Sept. 1743 [and] extends his principle of the 'vis viva potentialis laminae elasticae' to laminae of unequal elasticity, in which case \int E ds/R^2 is to be made a minimum. The... letter...in... April or May 1744... expresses his pleasure that Euler's results on the oscillations of laminae agree with his own."

"The celebrated work of Euler relating to... the Calculus of Variations appeared in 1744 under the title of Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. ...an appendix called Additamentum I. De Curvis Elasticis ...commences with a statement... shewing the theologico-metaphysical tendency... so characteristic of mathematical investigations in the 17th and 18th centuries. It was assumed that the universe was the most perfect conceivable, and hence arose the conception that its processes involved no waste, its 'action' was always the least required to effect a given purpose. ...Thus we find Maupertuis' extremely eccentric attempt at a principle of Least Action. ...[I]t is... probable that physicists have to thank this theological tendency in great part for the discovery of the modern principles of Least Action, of Least Constraint, and perhaps even of the Conservation of Energy."

"[S]tating that Daniel Bernoulli... had discovered... that the vis potentialis represented by \int ds/R^2 was a minimum for the elastic curve, Euler proceeds to discuss the inverse problem... The curve is to have a given length between two fixed points, to have given tangents at those points, and to render \int ds/R^2 a minimum... No attempt is made to shew why... By the aid of the principles of his book Euler arrives at the following equations where a, \alpha, \beta, \gamma are constants,dy = \frac{(\alpha + \beta x + \gamma x^2)}{\sqrt{a^4 - (\alpha + \beta x + \gamma x^2)^2}}from this we obtainds = \frac{a^2 dx}{\sqrt{a^4 - (\alpha + \beta x + \gamma x^2)^2}}"

"Euler gives... his investigation of the elastic curve in what he has just called an a priori manner. But this method is far inferior to that of James Bernoulli; for Euler does not attempt to estimate the forces of elasticity, but assumes that the moment of them at any point is inversely proportional to the radius of curvature: thus he... writes... an equation like... Poisson's Traite de Mecanique, Vol. I., without giving any of the reasoning by which Poisson obtains the equation."

"Euler... has hitherto considered the elasticity constant, but he will now suppose that it is variable... S, which is supposed a function of the arc s; \rho is the radius of curvature. He proceeds to find the curve which makes \int S ds/\rho^2 a minimum; and... finds for the differential equation of the required curve\alpha + \beta x -\gamma y = S/\rho where \alpha, \beta, \gamma are constants."

"Euler takes the case in which forces act at every point of the elastic curve; and he obtains an equation like the first volume of Poisson's Traiti de Mecanique."

"1757. Sur la force des colonnes, Mémoires de l'Académie de Berlin, Tom. XIII. 1759... is one of Euler's most important contributions to the theory of elasticity. The problem... is the discovery of the least force which will suffice to give any the least curvature to a column, when applied at one extremity parallel to its axis, the other extremity being fixed. Euler finds that the force must be at least = \pi^2 \cdot \frac{Ek^2}{a^2}, where a is the length of the column and Ek^2 is the 'moment of the spring' or the 'moment of stiffness of the column' (moment du ressort or moment de roideur)."

"If we consider a force F perpendicular to the axis of a beam (or lamina) so as to displace it from the position AC to AD, and \delta be the projection of D parallel to AC on a line through C perpendicular to AC, Euler finds by easy analysis D \delta = \frac{F\cdot a^3}{3\cdot Ek^2}, supposing the displacement to be small. This suggests to him a method of determining the 'moment of stiffness' Ek^2, and he makes various remarks on proposed experimental investigations. He then notes the curious distinction between forces acting parallel and perpendicular to a built-in rod at its free end; the latter, however small, produce a deflection, the former only when they exceed a certain magnitude. It is shewn that the force required to give curvature to a beam acting parallel to its axis would give it an immense deflection if acting perpendicularly."

"Euler deduces the equation for the curve assumed by the beam AC fixed but not built in at one end A and acted upon by a force P parallel to its axis. If RM be perpendicular to AC and y=RM, x = AM, he finds\frac{y}{\theta}\cdot \sqrt{\frac{P}{Ek^2}} = sin(x \sqrt{\frac{P}{Ek^2}}),where \theta = \angle RCM. Hence since y = 0, when x = a the length of the beam, a \sqrt{\frac{P}{Ek^2}} must at least = \pi, whence it follows that P must be at least = \pi^2 \cdot \frac{Ek^2}{a^2}. This paradox Euler seems unable to explain."

"If Q be the total weight of the beam the differential equationEk^2 ad^3 y + Pa (dx)^2 dy + Qx(dx)^2 dy = 0is obtained... This is reduced by a simple transformation to a special case of Riccati's equation, which is then solved on the supposition that \frac{Q}{P} is small. Euler obtains finally for the force P, for which the rod begins to bend, the expressionP = \pi^2 \cdot Ek^2/a^2 - Q \cdot (\pi^2 - 8)/2\pi^2;which shews that the minimum force is slightly reduced by taking the weight of the beam into consideration."

"Determinatio onerum quae columnae gestare valent. Examen insignis paradoxi in theoria columnarum occurrentis. De altitudine columnarum sub proprio pondere corruentium. [all in] Acta Academiae Petropolitanae [1778, 1780]. The first memoir... points out that vertical columns do not break under vertical pressure by mere crushing, but that flexure of the column will be found to precede rupture. ...[Euler] proposes to deduce a result which is now commonly in use... to find an expression connecting Ek^2 with the dimensions of the transverse section of the column. Euler finds Ek^2 = h \cdot \int x^2 ydx, where x and y... Euler appears however to treat the unaltered fibre or 'neutral line' without remark as the extreme fibre on the concave side of the section of the column made by the central plane of flexure. Thus for a column of rectangular section of dimensions b [with]in, and c perpendicular to the plane of flexure, he finds...Ek^2 = \frac{1}{3} b^3 ch, and the like method is used in the case of a circular section."

"Euler... calculate[s] the flexure which may be produced in a column by its own weight. If y be the horizontal displacement of a point on the column at a distance x from its vertex, the equation Ek^2 \cdot \frac{d^2y}{dx^2} + b^2 \int_0^y xdy = 0 is found, where the weight of unit volume of the column is unity and its section a square of side b. ...[I]f a be the altitude of the column and m = Ek^2/b^2, it is found that the least altitude for which the column will bend from its own weight is the least root of the equation,0 = \frac{1 \cdot a^3}{4! m} + \frac{1 \cdot 4 \cdot a^6}{7! m^2} - \frac{1 \cdot 4 \cdot 7 \cdot a^9}{10! m^2} + \frac{1 \cdot 4 \cdot 7 \cdot 10 \cdot a^{12}}{13! m^2} - \mathrm{etc.}Euler finds that this equation has no real root, and thus arrives at the paradoxical result, that however high a column may be it cannot be ruptured by its own weight. <!--(77-78.)p.45"

"P. S. Girard. Traite Analytique de la Resistance des solides, et des solides d'e'gale Resistance, Auquel on a joint une suite de nouvelles Experiences sur la force, et Velasticite specifique des Bois de Chine et de Sapin. Paris, 1798. ...This work very fitly closes the labours of the 18th century. It is the first practical treatise on Elasticity; and one of the first attempts to make searching experiments on the elastic properties of beams. It is not only valuable as containing the total knowledge of that day on the subject, but also by reason of an admirable historical introduction... The work appears to have been begun in 1787 and portions of it presented to the Academie in 1792. Its final publication was delayed till the experiments on elastic bodies, the results of which are here tabulated, were concluded at Havre. ...We are... considering the period of the French Revolution."

"The book... introduction is occupied with an historical retrospect of the work already accomplished in the field of elasticity... [and] concludes with an analysis of Girard's own work."

"The first section of Girard's treatise is concerned with the resistance of solids according to the hypotheses of Galilei, Leibniz and Mariotte. He notes Bernoulli's objections to the Mariotte-Leibniz theory; but remarks that physicists and geometricians have accepted this theory... [H]e thinks it probable that Galilei's hypothesis of non-extension of the fibres may hold for some bodies—stones and minerals—while the Mariotte-Leibniz theory is true for sinews, wood and all vegetable matters (cf. p. 6). As to Bernoulli's doubt with regard to the position of the neutral surface, Girard accepts Bernoulli's statement that the position of the axis of equilibrium is indifferent, and supposes accordingly that all the fibres extend themselves about the axis AC..."

"[Girard's] book forms... a most characteristic picture of the state of mathematical knowledge on the subject of elasticity at the time and marks the arrival of an epoch when science was to free itself from the tendency to introduce theologico-metaphysical theory in the place of the physical axiom deduced from the results of organised experience."

"General summary. As the general result of the work... previous to 1800... while a considerable number of particular problems had been solved by means of hypotheses more or less adapted to the individual case, there had as yet been no attempt to form general equations for the motion or equilibrium of an elastic solid. Of these problems the consideration of the elastic lamina by James Bernoulli, of the vibrating rod by Daniel Bernoulli and Euler, and of the equilibrium of springs and columns by Lagrange and Euler are the most important. The problem of a vibrating plate had been attempted, but with results which cannot be considered satisfactory."

"A semi-metaphysical hypothesis as to the nature of Elasticity was started by Descartes and extended by John Bernoulli and Euler. It is extremely unsatisfactory, but the attempt to found a valid dynamical theory by did not lead to any more definite results."

"In the appendix Mr. Pearson has carefully analysed the conflicting notations of different writers, and proposed a very convenient terminology and notation, which would save great trouble if universally adopted. He has also given an account of experiments carried out by Prof. Kennedy in his mechanical laboratory, which have an important bearing on the limitations of the truth of Hooke's law, or in the language of elasticity, the constancy of the ratio of stress to corresponding strain. The present volume is an indispensable hand-book of reference for the mathematician and the engineer, and in the editing and printing must be considered a very fitting tribute to the wonderful industry and application of its projector, the late Dr. Todhunter."

"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 summer of 1884... the Syndics... placed in my hands the manuscript of the late Dr Todhunter's History of Elasticity, in order that it might be edited and completed..."

"[I]t was not till I had advanced... into the work that I felt convinced that... the... writer's terminology and notation must be abandoned and a uniform terminology and notation adopted for the whole book... to be available for easy reference, and not merely of interest to the historical student."

"[T]he notation and terminology will be found fully discussed in Notes B—D of the Appendix, which I would ask the reader to examine before passing to the text."

"[C]onsistency in [notation and terminology] will be found after the middle of the chapter devoted to Poisson."

"The symbols and terms used in the manuscript are occasionally those of the original memoirs, occasionally those of Lamé or of Saint-Venant... the memoirs being of historical rather than scientific interest, and their language often the most characteristic part of their historical value."

"Dr Todhunter's manuscript consists of two distinct parts, the first contains a purely mathematical treatise on the theory of the 'perfect' elastic solid; the second a history of the theory of elasticity. The treatise based principally on the works of Lamé, Saint-Venant and Clebsch is yet to a great extent historical, [i.e.,] many paragraphs are composed of analyses of important memoirs."

"The changes I have made in that manuscript are of the following character; the introduction of a uniform terminology and notation, the correction of clerical and other obvious errors, the insertion of cross-references, the occasional introduction of a remark or of a footnote. The remarks are inclosed in square brackets. With this exception any article in this volume the number of which is not included in square brackets is due entirely to Dr Todhunter."

"I... regret that I have not devoted special chapters to such elasticians as Hodgkinson, [Guillaume] Wertheim and F. E. Neumann; in the latter case the regret is deepened by the recent publication of his lectures on elasticity."

"I may appear to have exceeded the duty of an editor. For all the Articles in this volume whose numbers are enclosed in square brackets I am alone responsible, as well as for the corresponding footnotes, and the Appendix with which the volume concludes."

"The principle which has guided me throughout the additions I have made has been to make the work, so far as it lay in my power, a standard work of reference for its own branch of science."

"The use of a work of this kind is twofold. It forms on the one hand the history of a peculiar phase of intellectual development, worth studying for the many side lights it throws on general human progress. On the other hand it serves as a guide to the investigator in what has been done, and what ought to be done. In this latter respect the individualism of modern science has not infrequently led to a great waste of power; the same bit of work has been repeated in different countries at different times, owing to the absence of such histories as Dr Todhunter set himself to write. ...the various Jahrbücher and Fortschritte now reduce the possibility of this repetition, but besides their frequent insufficiency they are at best but indices to the work of the last few years; an enormous amount of matter is practically stored out of sight in the Transactions and Journals of the last century and of the first half of the present century."

"It would be a great aid to science, if, at any rate, the innumerable mathematical journals could be to a great extent specialised, so that we might look to any one of them for a special class of memoir. ...the would-be researcher either wastes much time in learning the history of his subject, or else works away regardless of earlier investigators. The latter course has been singularly prevalent with even some firstclass British and French mathematicians."

"Keeping the twofold object of this work in view I have endeavoured to give it completeness (1) as a history of developement, (2) as a guide to what has been accomplished."

"Taking the first chapter of this History the author has discussed the important memoirs of James Bernoulli and some of those due to Euler. The whole early history of our subject is however so intimately connected with the names of Galilei, Hooke, Mariotte and Leibniz, that I have introduced some account of their work."

"The labours of Lagrange and Riccati also required some recognition... [of] interest, whether judged from the special standpoint of the elastician or from the wider footing of insight into the growth of human ideas."

"With a similar aim I have introduced throughout the volume a number of memoirs having purely historical value which had escaped Dr Todhunter's notice."

"I have inserted... memoirs of mathematical value, omitted [by Todhunter] apparently by pure accident. For example all the memoirs of F. E. Neumann, the second memoir of Duhamel, those of Blanchet etc. I cannot hope that the work is complete in this respect even now, but I trust that nothing of equal importance has escaped..."

"My greatest difficulty arose with regard to the rigid line which Dr Todhunter had attempted to draw between mathematical and physical memoirs. Thus while including an account of Clausius' memoir of 1849, he had omitted Weber's of 1835, yet the consideration of the former demands the inclusion of the latter..."

"There has been far too much invention of 'solvable problems' by the mathematical elastician; far too much neglect of the physical and technical problems which have been crying out for solution. Much of the ingenuity which has been spent on the ideal body of 'perfect' elasticity ideally loaded, might I believe have wrought miracles in the fields of physical and technical elasticity, where pressing practical problems remain in abundance unsolved. I have endeavoured... to abrogate this divorce between mathematical elasticity on the one hand, and physical and technical elasticity on the other. With this aim in view I have introduced the general conclusions of a considerable body of physical and technical memoirs, in the hope that by doing so I may bring the mathematician closer to the physicist and both to the practical engineer. I trust that in doing so I have rendered this History of value to a wider range of readers, and so increased the usefulness of Dr Todhunter's many years of patient historical research on the more purely mathematical side of elasticity. In this matter I have kept before me the labours of M. de Saint-Venant as a true guide to the functions of the ideal elastician."

"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."