elasticity-physics

"The assumption that the varies as the curvature is the basis of the 'Euler-Bernoulli' theory of flexure. This was developed in memoirs by James Bernoulli (1705) D. Bernoulli (1742), L. Euler (1744)."

"We consider the case of a horizontal rod or beam slightly bent by vertical forces applied to it. The state of strain is no longer of the simple character appertaining to pure flexure; in particular there will be a relative shearing of adjacent cross-sections, and also a warping of the sections so that these do not remain accurately plane. We shall assume, however, that the additional strains thus introduced are on the whole negligible, and consequently that the bending moment is connected with the curvature of the axis..."

"Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by , was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the s. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of s, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions."

"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."

"The assumptions in the design of reinforced concrete beams are those of the ordinary beam theory, namely: the Bernoulli-Euler theory of flexure. The fundamental premise is that a plane section before bending, remains a plane section after bending, with the further assumption that , i.e., the stress is proportional to the strain, is true. Although the brilliant researches of Barre de St. Venant, have shown that plane sections do not remain plane during bending, the error becomes appreciable when the ratio of depth of beam to span exceeds one-fifth. Since for such ratios, stresses, other than those induced by , usually govern the required reinforcement and depth of beam e.g. the unit shear and adhesion, these assumptions of plane sections may be taken as valid, so long as the stresses induced by the bending moment govern the required depths and amounts of steel reinforcement. The concrete is assumed to take no tension."

"[C]alculations... based on the simple theory of bending... are approximate only. While the simple (or Bernoulli-Euler) theory gives the deflections due to the bending moment with sufficient accuracy, the portion of the total deflection which is due to shearing cannot generally be estimated with equal accuracy from the distribution of shear stress... It becomes desirable, then, to check the results by those given in the more complex theory of St. Venant... if a very accurate estimate of shearing deflection is required. In a great number of practical cases, however, the deflection due to shearing is negligible in comparison with that caused by the bending moment."

"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... ."

"[W]hile the mathematicians were beginning to struggle with the problems of elasticity, a number of practical experiments were being made on the flexure and rupture of beams, the results of which were of material assistance to the theorists."

"The method of James Bernoulli with improvements, has been substantially adopted by other writers. The English reader may consult the earlier editions of Whewell's Mechanics. Poisson says in his Traité de Mécanique... Jacques Bernoulli a déterminé, le premier, la figure de la lame élastique en équilibre, d'après des considérations que nous allons développer, . . .[Jacques Bernoulli has determined, the first, the figure of the elastic blade in equilibrium, according to considerations that we will develop...]"

"Petris van Musschenbroek: Introductio ad cohaerentiam corporum firmorum.... commences at of the author's Physicae experimentales et geometricae Dissertationes. Lugduni 1729. It was held in high repute even to the end of the 18th century. ...[The] historical preface,... has been largely drawn upon by Girard. [Van Musschenbroek] describes the various theories which have been started to explain cohesion, and rejects successively that of the pressure of the air and that of a subtle medium. ...He laughs at Bacon 's explanation of elasticity, and another metaphysical hypothesis he terms abracadabra. ...[H]e falls back... upon Newton's thirty-first Query... and would explain the matter by vires internae [internal forces]. Musschenbroek assumes... we may determine them in each case by experiment. ...The source of elasticity is a vis interna attrahens... drawn directly from Newton's Optics."

"The fourth Lemma... may be readily understood by reference to Varignon's memoir. ...Varignon supposed the neutral surface to pass through... the so-called 'axis of equilibrium'... James Bernoulli... recognises the difficulty of determining the fibres which are neither extended nor compressed, but he comes to the conclusion that the same force applied at the extremity of the same lever will produce the same effect, whether all the fibres are extended, all compressed or part extended and part compressed about the axis of equilibrium. In other words the position of the axis of equilibrium is indifferent. This result is expressed by the fourth Lemma and is of course inadmissible."

"James Bernoulli next takes a problem which he enunciates thus: "Trouver combien il faut plus de force pour rompre une poutre directement, c'est-à-dire en la tirant suivant sa longueur, que pour la rompre transversalement." [Find out how much more force is needed to break a beam directly... by pulling it along its length in order to break it transversely.] The investigation depends on the fourth Lemma, and is consequently not satisfactory."

"After using arguments from capillarity to confirm these remarks he continues:Now the small particles of matter may cohere by the strongest attractions, and compose bigger particles of weaker virtue; and many of these may cohere and compose bigger particles, whose virtue is still weaker; and so on for divers successions, until the progression end in the biggest particles, on which the operations in chemistry, and the colours of natural bodies depend; and which by adhering, compose bodies of a sensible magnitude. If the body is compact, and bends or yields inward to pression without any sliding of its parts, it is Hard and Elastick, returning to its figure with a force rising from the mutual attractions of its parts."

"This seems to be Newton's only contribution to the subject of Elasticity, beyond the paragraph of the Principia on the collision of elastic bodies."

"Mariotte seems to have been the earliest investigator who applied anything corresponding to the elasticity of Hooke to the fibres of the beam in Galilei's problem. ...[H]is Traité du mouvement des eaux, Paris 1686... shows that Galilei's theory does not accord with experience. He remarks that some of the fibres of the beam extend before rupture, while others again are compressed. He assumes however without the least attempt at proof ("on peut concevoir" [we can conceive]) that half the fibres are compressed, half extended."

"The modern theory of elasticity may be considered to have its birth in 1821, when Navier first gave the equations for the equilibrium and motion of elastic solids, but some of the problems which belong to this theory had previously been solved or discussed on special principles, and to understand the growth of our modern conceptions it is needful to investigate the work of the seventeenth and eighteenth centuries."

"G. W. Leibniz: Demonstrationes novae de Resistentiâ solidorum. Acta Eruditorum Lipsiae July 1684. The stir created by Mariotte's experiments... seem to have brought the German philosopher into the field. He treats the subject in a rather ex cathedrâ fashion, as if his opinion would finally settle the matter. He examines the hypotheses of Galilei and Mariotte, and finding that there is always flexure before rupture, he concludes that the fibres are really extensible. Their resistance is, he states, in proportion to their extension. ...[i.e.,] he applies " Hooke's Law" to the individual fibres. As to the application of his results to special problems, he will leave that to those who have leisure for such matters. The hypothesis... is usually termed by the writers of this period the Mariotte-Leibniz theory."

"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."

"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..."

"My colleague, Professor A. B. W. Kennedy, has continually placed at my disposal the results not only of special experiments, but of his wide practical experience. The curves figured in the Appendix, as well as a variety of practical and technical remarks scattered throughout the volume I owe entirely to him; beyond this it is difficult for me to fitly acknowledge what I have learnt from mere contact with a mind so thoroughly imbued with the concepts of physical and technical elasticity."

"A beam (ABCD) being built horizontally into a wall (at AB) and strained by its own or an applied weight (E), to find the breaking force upon a section perpendicular to its axis. This problem is always associated... with Galilei's name, and we shall call it... Galilei's Problem. The 'base of fracture' being defined as the section of the beam where it is built into the wall; we have the following results :— (i) The resistances of the bases of fracture of similar prismatic beams are as the squares of their corresponding dimensions. In this case the beams are supposed loaded at the free end till the base of fracture is ruptured; the weights of the beams are neglected. (ii) Among an infinite number of homogeneous and similar beams there is only one, of which the weight is exactly in equilibrium with the resistance of the base of fracture. All others, if of a greater length will break,—if of a less length will have a superfluous resistance in their base of fracture."

"The modern expression of the six components of stress as linear functions of the strain components may perhaps he physically regarded as a generalised form of ."

"In Galilei's hypothesis of inextensible fibres u is supposed constant = r and the resistance of the base of fracture becomesr \int ydxdy = \frac{r}{2} \cdot \int y^2dx.On the supposition that the fibres are extensible we ought to consider their extension by finding what is now termed the neutral line or surface. Varignon however, and he is followed by later writers, assumes that the fibres in the base ACLN are not extended; and that the extension of the fibre through H' varies as DH, in other words he makes the curve GK a straight line passing through D. Hence if r' be the resistance of the fibre at B, and DB = a, the resistance of the fibre at H = r'y/a or the resistance of the base of fracture on this hypothesis becomes\frac{r'}{3a}\int y^3dxThis resistance in the case of a rectangular beam of breadth b and height a becomes on the two hypotheses\frac{ra^2b}{2} and \frac{r'a^2b}{3}...his results are practically vitiated when applying the true ( Leibniz-Mariotte) theory by his assumption of the position of the neutral surface, but in this error he is followed by so great a mathematician as Euler himself."

"Three Lemmas which present no difficulty are given and demonstrated [by James Bernoulli]: I. Des Fibres de même matière et de même largeur, ou épaisseur, tirées ou pressées par la même force, s'étendent ou se compriment proportionellement à leurs longueurs. [Fibers of the same material and of the same width, or thickness, drawn or pressed by the same force, extend or compress proportionally to their lengths.] II. Des Fibres homogènes et de même longueur, mais de différentes largeurs ou épaisseurs, s'étendent ou se compriment également par des forces proportionelles à leurs largeurs. [Fibers homogeneous and of the same length, but of different widths or thicknesses, extend or are also compressed by forces proportional to their widths.] III. Des Fibres homogènes de même longueur et largeur, mais chargées de différens poids, ne s'étendent ni se compriment pas proportionellement à ces poids; mais l'extension ou la compression causée par le plus grand poids, est à l'extension ou à la compression causée par le plus petit, en moindre raison que ce poids—là n'est à celui—ci. [Homogeneous fibers of the same length and width, but charged with different weights, neither extend nor compress proportionally to these weights; but the extension or the compression caused by the greatest weight, is to the extension or to the compression caused by the smaller, in less reason...]"

"Saint-Venant remarks in his memoir on the Flexure of Prisms in Liouville's Journal, 1856: On s'étonne de voir, vingt ans plus tard, un grand géomètre, auteur de la première théorie des courbes élastiques, Jacques Bernoulli tout en admettant aussi les compressions et présentant même leur considération comme étant de lui commettre sous une autre forme, précisement la même méprise du simple au double que Mariotte dans l'évaluation du moment des résistances ce qui le conduit même à affirmer que la position attribuée à l'axe de rotation est tout à fait indifférente. [It is surprising to see, twenty years later, a great geometer, author of the first theory of elastic curves, Jacques Bernoulli... commit precisely the same mistake of... Mariotte in the evaluation of moment of resistance which leads him... to assert that the position attributed to the axis of rotation is entirely indifferent.]"

"Bernoulli... rejects the Mariotte-Leibniz hypothesis or the application of Hooke's law to the extension of the fibres. He introduces rather an idle argument against [it], and quotes an experiment of his own which disagrees with Hooke's Ut tensio, sic vis."

"Sir Isaac Newton : Optics or a Treatise of the Reflections, Refractions and Colours of Light. 1717. ...The Query [XXXIst, termed 'Elective Attractions,'] commences by suggesting that the attractive powers of small particles of bodies may be capable of producing the great part of the phenomena of nature:—For it is well known that bodies act one upon another by the attractions of gravity, magnetism and electricity; and these instances shew the tenor and course of nature, and make it not improbable, but that there may be more attractive powers than these. For nature is very consonant and conformable to herself. ... The parts of all homogeneal hard bodies, which fully touch one another, stick together very strongly. And for explaining how this may be, some have invented hooked atoms, which is begging the question; and others tell us, that bodies are glued together by Rest: that is, by an occult quality, or rather by nothing: and others, that they stick together by conspiring motions, that is by relative Rest among themselves. I had rather infer from their cohesion, that their particles attract one another by some force, which in immediate contact is exceeding strong, at small distances performs the chemical operations above-mentioned, and reaches not far from the particles with any sensible effect."

"Newton supposes all bodies to be composed of hard particles, and these are heaped up together and scarce touch in more than a few points.And how such very hard particles, which are only laid together, and touch only in a few points can stick together, and that so firmly as they do, without the assistance of something which causes them to be attracted or pressed towards one another, is very difficult to conceive."

"The conception of repulsive forces is then introduced [by Newton] to explain the expansion of gases.Which vast contraction and expansion seems unintelligible, by feigning the particles of air to be springy and ramous, or rolled up like hoops, or by any other means than a Repulsive power. And thus Nature will be very conformable to herself, and very simple; performing all the great motions of the heavenly bodies by the attraction of gravity, which intercedes those bodies; and almost all the small ones of their particles, by some other Attractive and Repelling powers."

"A suggestive paragraph... occurs... which is sometimes not sufficiently remembered when gravitation is spoken of as a cause :—These principles—i.e. of attraction and repulsion—I consider not as occult qualities, supposed to result from the specifick forms of things, but as general laws of Nature, by which the things themselves are formed; their truth appearing to us by phenomena, though their causes be not yet discovered."

"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."

"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 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."

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

"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 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."

"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..."

"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."

"To the late M. Barré de Saint-Venant I am indebted for the loan of several works, for a variety of references and facts bearing on the history of elasticity, as well as for a revision..."

"Galileo Galilei['s] second dialogue of the Discorsi e Dimostrazioni matematiche, Leiden 1638... both from its contents and form is of great historical interest. It not only gave the impulse but determined the direction of all the inquiries concerning the rupture and strength of beams, with which the physicists and mathematicians for the next century principally busied themselves."

"Galilei gives 17 propositions with regard to the fracture of rods, beams and hollow cylinders. ...[H]e supposed the fibres of a strained beam to be inextensible. There are two problems... discussed... which form the starting points of many later memoirs. They are the following:"

"The discovery apparently of the modern conception of elasticity seems due to Robert Hooke, who in his work De potentiâ restitutiva, London 1678, states that 18 years before... he had first found out the theory of springs, but had omitted to publish it because he was anxious to obtain a patent for a particular application of it. He continues:— About three years since His Majesty was pleased to see the Experiment that made out this theory tried at White-Hall, as also my Spring Watch. About two years since I printed this Theory in an Anagram at the end of my Book of the Descriptions of s, viz. ceiiinosssttuu, id est, Ut Tensio sic vis; That is, The Power of any spring is in the same proportion with the Tension thereof."

"By 'spring' Hooke does not merely denote a spiral wire, or a bent rod of metal or wood, but any "springy body" whatever. Thus after describing his experiments he writes: From all which it is very evident that the Rule or Law of Nature in every springing body is, that the force or power thereof to restore it self to its natural position is always proportionate to the Distance or space it is removed therefrom, whether it be by rarefaction, or separation of its parts the one from the other, or by a Condensation, or crowding of those parts nearer together. Nor is it observable in these bodies only, but in all other springy bodies whatsoever, whether Metal, Wood, Stones, baked Earths, Hair, Horns, Silk, Bones, Sinews, Glass and the like. Respect being had to the particular figures of the bodies bended, and to the advantageous or disadvantageous ways of bending them."

"Let ABCNML be a beam built into a vertical wall at the section ABC, and supposed to consist of a number of parallel fibres perpendicular to the wall... and equal to AN in length. Let H' be a point on the 'base of fracture,' and H'E [which is perpendicular to AC] = y, AE= x. Then if a weight Q be attached by means of a pulley to the extremity of the beam, and be supposed to produce a uniform horizontal force over the whole section NML, \; Q = r \cdot \int ydx where r is the resistance of a fibre of unit sectional area and the integration is to extend over the whole base of fracture. Q is by later writers termed the absolute resistance and is given by the above formula. Now suppose the beam to be acted upon at its extremity by a vertical force P instead of the horizontal force Q. All the fibres in a horizontal line through H' will have equal resistance, this may be measured by a line HK drawn through H in any fixed direction where H is the point of intersection of the horizontal line through H and the central vertical BD of the base. As H moves from B to D, K will trace out a curve GK which gives the resistance of the corresponding fibres. Take moments for the equilibrium of the beam about ACP \cdot l = \iint uydxdywhere l = length of the beam DT and u = HK."

"This quantity \iint uydxdy was termed the relative resistance of the beam or the resistance of the base of fracture. ...it is necessary to know u before we can make use of it. He then proceeds to apply it to Galilei's and the Mariotte-Leibniz hypotheses."

"The first work of genuine mathematical value on our subject is clue to James Bernoulli... Véritable hypothèse de la résistance des Solides, avec la démonstration de la Courbure des Corps qui font ressort... 12th of March 1705... begins by brief notices of what had been already done with respect to the problem by Galilei, Leibniz, and Mariotte; James Bernoulli claims for himself that he first introduced the consideration of the compression of parts of the body, whereas previous writers had paid attention to the extension alone."

"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."