APPENDIX. NOTE A. ADDENDA. (1) The Stress-Strain Curve. Arts. 18 and 979. 1692. James Bernoulli. Curvatura Laminae Elasticae. Acta Eruditorum, Leipzig, 1694, pp. 262-276. On p. 265 Bernoulli writes: Esto Spatium rectilineum sive curvilineum quodvis ABC, cujus abscissae AE vires tendentes, ordinatae EF tensiones repraesentent etc. This curve he terms the curva tensionum or linea tensionum. Bernoulli might thus be considered to have introduced a graphical method of representing the stress-strain relation. At the same time it will be seen by consulting the original memoir that Bernoulli's linea tensionum does not represent the curve obtained by measuring the strains produced in the same rod by a continually increasing stress. This seems to me to have been first done by Poncelet. (2) The Coulomb-Gerstner Law. Arts. 119 and 806, footnote. 1784. Coulomb. Recherches théoriques et expérimentales sur la force de torsion et sur l'élasticité des fils de métal. Histoire de l'Académie des Sciences, année 1784, Paris, 1787, pp. 229-269. This memoir is reprinted Tome 1. pp. 63-103 of the Collection de Mémoires relatifs à la Physique publiés par la Société Française de Physique, Paris, 1884. Dr Todhunter has referred to it in Art. 119. We may note the second section (pp. 255-269) of the memoir in this place. It occupies pp. 90103 of the reprint and is entitled: De l'altération de la force élastique dans les torsions des fils de métal. Théorie de la cohérence et de l'élasticité. In this section Coulomb brings out clearly (i) that the absolute strength of a material depends upon the working or treatment it has received (la force des métaux varie suivant le degré d'écrouissement et de recuit); (ii) that in the case of set, the set-slide produced by torsion is at first for very small sets proportional to the total slide and thus to the elastic slide, but that it very soon begins to increase in a much greater ratio; (iii) that the slide-modulus (Coulomb speaks of the réaction de torsion) remains almost the same after any slide-sets; (iv) that the elastic limit (at least for the case of torsion) can be extended by giving the material a set, thus by subjecting a wire to great torsional set a state of ease can be produced almost as extended as if the wire had been annealed. Coulomb also notes that the resistance of the air had very little effect in diminishing the amplitude of the oscillations of his apparatus, and he apparently attributes the decrease in amplitude to something akin to 'fatigue of elasticity'. 1 Coulomb distinguishes elasticity and cohesion as absolutely different properties. Thus the cohesion can be much altered by working or other treatment. It would appear that the elastic limit is altered when the cohesion or absolute strength is altered'; scarcely however, as Coulomb apparently suggests, in the same ratio (§ XXXI.). The elastic constants however remain the same: see our Art. 806. This independence of the elasticity of the cohesion was confirmed by flexure experiments for the stretch-modulus (§ XXXIII.). It may be noted that Coulomb uses the terms glissement and glisser for set-slide and not for elastic slide. He appears to hold that the difference in the cohesion of the same material in different states depends upon its capacity for receiving set-slide. If its parts cannot slide on each other, it is brittle; if they can, it is ductile or malleable. It will be noted that the several suggestive points of this memoir remained for many years unregarded, till they were again rediscovered by Gerstner and Hodgkinson. (3) Resilience. Arts. 136, 993 and 999. (a) J. A. Borellus. Liber de vi percussionis, Bologna, 1667. Another edition of this work entitled: De vi percussionis et motionibus naturalibus a gravitate pendentibus (Editio prima Belgica, priori Italicâ multo correctior et auctior, etc.), appeared 1686, Lugduni Batavorum. There are two chapters in this work (Caput XVIII. and Caput XIX.) entitled: 1 The ratio elastic limit as the cohesive limit increases, but with many materials in a worked condition (e. g. tempered steel) this is not true. Quomodò in flexibilibus corporibus impetus impressus retardetur aut extinguatur (pp. 106-112 of the later edition). Qua ratione in corporibus flexibilibus resilientibus motus contrarii se mutuò destruant, renoventurque (pp. 112-116 of the later edition). These chapters at least by their titles suggest a consideration of the problem of resilience, and the examination of the figures on Plate III., opposite p. 106, suggests still more strongly that something of value might be found in them. Beyond giving, however, the name resilience, probably for the first time, to a number of problems now classed under that name, the work really contributes nothing to our subject, being composed of a number of general and extremely vague propositions. (b) 1807. Young. A Course of Lectures on Natural Philosophy and the Mechanical Arts. Young was, I believe, the first to introduce into English the term resilience, and to state the general theorem that: The resilience of a prismatic beam resisting a transverse impulse is simply proportional to the bulk or weight of the beam. The statement of this general proposition occurs on p. 147, Vol. 1. of his Lectures on Natural Philosophy. On p. 50 of Vol. II. he returns to the matter with the following definition and theorem : The resilience of a beam may be considered as proportional to the height from which a given body must fall to break it. The resilience of prismatic beams is simply as their bulk. This theorem he proves in the following characteristic fashion: The space through which the force or stiffness of a beam acts, in generating or destroying motion, is determined by the curvature that it will bear without breaking; and this curvature is inversely as the depth; consequently, the depression will be as the square of the length directly, and as the depth inversely: but the force in similar parts of the spaces to be described is everywhere as the strength, or as the square of the depth directly, and as the length inversely: therefore the joint ratio of the spaces and the forces is the ratio of the products of the length by the depth; but this ratio is that of the squares of the velocities generated or destroyed, or of the heights from which a body must fall to acquire these velocities. And if the breadth vary, the force will obviously vary in the same ratio; therefore the resilience will be in the joint ratio of the length, breadth and depth. It will be observed that Young is here speaking of cohesive resilience, which must be distinguished from elastic resilience. The latter has of course greater practical importance. Compare Note E, (b). (4) Fourier. Art. 207. An account of a memoir by Fourier upon the vibrations of flexible and extensible surfaces, and of elastic plates, will be found on pp. 258264 of the Histoire de l'Académie des Sciences in the Analyse des Travaux...pendant l'Année 1822. The Analyse will be found attached to Tome v. of the Mémoires (1821 and 1822) published in 1826. Delambre seems to have analysed this memoir, which never appears to have been published as a whole. It apparently contained solutions of the linear partial differential equations satisfied by the above forms of vibrations of the kind given for similar equations in Fourier's Théorie de la Chaleur, i.e. solutions in the form of periodic series and of definite integrals. (5) Addendum to Chapter III. H. F. Eisenbach. Versuch einer neuen Theorie der Kohäsionskraft und der damit zusammenhängenden Erscheinungen, Tübingen, 1827. I have added a reference in the Addenda to this book as its title might lead a reader to believe something of value was contained in Eisenbach's theory. This would undoubtedly be the case were we to accept the author's own estimate of his discovery, the history of which he narrates in some 18 pages. The 'new theory of cohesion' consists in the hypothesis that the law of cohesion is based on a central intermolecular force which can be expanded in inverse powers of the square of the molecular distance. On this hypothesis Eisenbach attempts, with the crudest mixture of mathematical and physical absurdities, worthy of the Père Mazière, to explain cohesive, elastic and chemical phenomena. He speaks of the memoirs of Euler, Lagrange and Laplace as vortreffliche Vorarbeiten for his own great principle, although he criticises somewhat severely Euler's memoir of 1778: see our Art. 74. Those who are interested in the history of pseudo-science, in the paradoxes and self-admiration of circle-squarers, perpetual-motion seekers, gold-resolvers, and the innumerable 'Grübler die so lange über einem Trugschlusse brüten, bis er zur fixen Idee wird und als Wahrheit erscheint,' will find much amusement and instruction in Eisenbach's sections: Eine Widerlegung aller bisherigen Kohäsionstheorien and Kurze Geschichte des Ganges meiner Erfindung. Englishmen should remark that it was on board the 'Emilie' in the London Docks that the possibility of this great principle 'stepped like a flash of lightning before the soul' of this second, but sadly disregarded Newton. (6) Slide-Glissement. Arts. 120, 143, 279 and 726. I have shewn in Art. 120 that Coulomb had formed the important conceptions of 'lateral adhesion' and of 'sliding strain.' In the paragraph there quoted Coulomb talks of a stress tending à faire couler la partie supérieure du pilier sur le plan incliné par lequel il touche la partie inférieure. In Coulomb's memoir of 1773 (see our Art. 115) we have a section, pp. 348-349, on Cohésion. Here Coulomb describes an experiment on shearing force: J'ai voulu voir si en rompant un solide de pierre, par une force dirigée suivant le plan de rupture, il fallait employer le même poids que pour le rompre, comme dans l'expérience précédente, par un effort perpendiculaire à ce plan. He found that the shearing load must be slightly greater than the tractive load, but the difference was so little (of the total load) that he neglects it in the theory which follows. On the uni-constant hypothesis the limit of shearing load should be of the limit of tractive load. A little later on in the memoir (p. 353) Coulomb uses the expression tendre à glisser; for exactly the same conception as in the former paragraph he used tendre à faire couler. It will thus be seen that although Coulomb's theory is unsatisfactory he had still formed a clear conception of slide and shear. In Art. 143 we have noted that Young in 1807 drew attention to the phenomena of 'lateral adhesion,' or as he terms it detrusion. It was however Vicat who first insisted on the mechanical importance of this form of strain and the accompanying stress. In his memoir of 1831 he defines shear (force transverse) in terms of the strain which it tends to produce. He uses the word glisser: see our Art. 726. Vicat's remarks did not escape Navier, who in the second edition of his Leçons (see our Art. 279), which appeared in 1833, after describing the nature of shear thus defines the slide-modulus : Un coefficient spécifique représentant la résistance du corps à un glissement d'une partie sur l'autre dans le plan de la section transversale (§ 152). The merit of practically introducing slide and shear into the ordinary theory of beams rests, as we have seen in Arts. 1564-1582, with Saint-Venant. (7) Notation for the six stresses. Art. 610, footnote. In the table of notations I have attributed to Kirchhoff the notation: |