Abstract
The first new name associated with elasticity was that of Fourier. An 1818 note shows how a solution to the plate equation could be obtained through an approach uniquely associated with his name: through the representation of plate motion in the form of a summation over the trigonometric functions, sine and cosine. His focus was on solution technique, not on a demonstration or generalization of the plate equation. The particular problem he solved also avoided a deficiency in theory that existed at that time, namely, the lack of an accepted set of boundary conditions. Since he intended to describe the response of an infinite plate to a quite arbitrary initial disturbance, the need for knowing conditions at an edge did not arise.1
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Chapter Nine: Emergence of a Theory
Fourier, J.: 1818, ‘Note relative aux vibrations des surfaces élastiques…’, Bull. Soc. Philom. de Paris, pp. 129–136. Poisson wrote a short note attacking the same problem employing a more traditional method to obtain a solution. He showed that Fourier’s solution was contained in his own. Poisson, S. D.: 1818, ‘Sur l’intégrale de l’équation relative aux vibrations des plaques élastiques’, Bull. Soc. Philom. de Paris, pp. 125–128.
A copy of Navier’s Mémoire sur la flexion des plans élastiques is held at the Archives, Bibliothèque, Ecole Nationale des Ponts et Chaussées, Paris. Navier, at that time, was not a member of the Royal Academy. How widely he distributed his paper is unknown.
In Timoshenko, S.: 1959, Theory of Plates and Shells, (2nd ed.), McGraw-Hill, New York. The author, in one section of this modern treatise, has reproduced and embellished Navier’s analyses of several phenomena, including the deflection of a uniformly loaded, simply supported, rectangular plate and the deflection of the same plate subjected to a concentrated load.
See Timoshenko S.: 1953, History of the Strength of Materials, McGraw-Hill, New York.
Todhunter, I., and Pearson, K.: 1886, A History of the Theory of Elasticity and of the Strength of Materials, At the University Press, Cambridge. An extract of Navier’s memoir was published in the Bull. Soc. Philom. de Paris (1823), pp. 177–181. His complete analysis was not published until 1827. Navier’s approach yields a one-constant theory of elasticity. We know today that two independent constants are required to specify the material properties of an isotropic, elastic solid. The question of how many independent constants were required to define the character of an elastic body was addressed by a variety of nineteenth-century scientists including Green. Lamé, Stokes, Clausius, Kirchoff, and St. Venant, as well as Cauchy and Poisson.
P. V., vol. 7, p. 296. An extract of Navier’s analysis of the motion of fluids appeared in the Bull. Soc. Philom. de Paris (1825): pp. 49–52. The complete memoir is found in the Mem. Acad. Sci. 6, pp. 389–440.
P.V., vol. 7. p. 370.
Ibid., p. 371.
Cauchy, 1823, ‘Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non-élastiques’, Bull. Soc. Philom. de Paris, pp. 9–13. At the end of this abstract Cauchy promised, in a future memoir, to analyze the behavior of elastic plates and beams. As noted in the text, Navier’s memoir on plates to which Cauchy refers — “a memoir on plates published the 14th August 1820…”— had appeared only in the form of a privately distributed lithograph edition.
Cauchy toyed with two types of possible stress-strain relations. First he assumed that the three principal stresses at a point were respectively proportional to the three principal strains, e.g. σI = k∈I, σII = k∈II, σIII = k∈III. This yields a one-constant, (k), theory of elasticity similar to, but not the same as, that derived by Navier. Second, in a modification of this hypothesis, he assumed stress depended upon strain in the following way \( \begin{gathered} {\sigma_I} = k{ \in_I} + Kv \hfill \\ {\sigma_{{II}}} = k{ \in_{{II}}} + Kv \hfill \\ {\sigma_{{III}}} = k{ \in_{{III}}} + Kv \hfill \\ \end{gathered} \) where K is a second elastic constant and ν, the volume dilatation. He noted that setting K = k/2 yields Navier’s equations of equilibrium in terms of displacement. The details of Cauchy’s analysis appear in Cauchy, A.: 1828, Exercises de mathématiques, année 1828, in vol. 8 of Oeuvres Complètes, Sér. 2, 13 vols., Gauther-Villars, Paris, 1897–1958. There, in a section ‘Sur les équations qui expriment les conditions d’équilibre ou les lois du mouvement intérieur d’un corps solide, élastique ou non-élastique’, Cauchy states that most of the equations derived in this section had been drawn from his memoir read to the Academy on 30 September, 1822, (p. 215). For an evaluation of Cauchy’s work see Truesdell, C: 1961, ‘Stages in the Development of the Concept of Stress’, in Problems of Continuum Mechanics, Muskhelishvili Anniversary Volume, (English edn.), Soc. for Indust. and Appl. Math., Philadelphia.
See also by the same author, Truesdell, C: 1962, The Creation and Unfolding of the Concept of Stress’, Essays in the History of Mechanics, Springer-Verlag, Berlin.
Navier, C.: 1823, ‘Observations communiquées par M. Navier, à l’occasion du Mémoire de M. Cauchy,” Bull. Soc. Philom. de Paris, pp. 36–37.
Navier, C: 1823, ‘Extrait des recherches sur la flexion des plans élastiques’, Bull. Soc. Philom. de Paris, pp. 92–102. Navier’s description of how he derived the plate equation in this abstract of his work differs significantly from his description in his lithographed, 1820 memoire. The extract reads as though Navier had deduced his expression for the elastic moment from a molecular model. In his 1820 memoir he worked without making any reference to sensible forces at insensible distances. A summary of Navier’s ‘Sur les lois de l’équilibre et du mouvement des corps solides élastiques’ was published in Bull. Soc. Philom. de Paris (1823) 177–181.
P. V., vol. 8, p. 9 (Session of 19 January, 1824).
Letter from Fourier to Sophie Germain, undated but obviously written shortly after Navier had read his memoir to the Academy, held at the Bibliotèque Nationale, MS. Fr. (Nouv. Acq.) 4073. Also in Henry, C.: 1879, ‘Les manuscrits de Sophie Germain’, Revue Phil. 8, p. 630.
Navier, ‘Extrait’, op. cit., p. 93.
Letter from Fourier to Sophie Germain dated 15 March, 1824. Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 320. The postscript indicates that this letter was written the Friday after the presentation of her memoir.
The appearance in print of Germain, S.: 1880, Mémoire sur l’emploi de lépaisseur dans la théorie des surfaces élastiques, Gauthier-Villars, Paris, followed Stupuy’s publication of her Oeuvres philosophiques in 1879. The cover page of Sophie Germain’s manuscript, which until 1879 had been closeted away with Prony’s papers in the Archives of the Ecole des Ponts et des Chaussées, indicates that Poisson alone had perused her work. Her manuscript is now held at the Archives of the Academy of Sciences, Paris.
Germain, S.: 1826, Remarques sur la nature, les bornes et l’étendue de la question des surfaces élastiques, Huzard-Courcier, Paris. Here Sophie Germain provides a geometrical interpretation of her expression for the elastic force of a plate; this force is simply measured by the radius of curvature of an equivalent sphere, the radius of this sphere being equal to the mean curvature at a point in the deformed plate. In her introduction she noted that her previous memoir had been presented to the Academy two years before but Prony and Poisson had not, as yet, made their report. She promised to publish this earlier treatise when circumstances allowed.
Letter from Fourier to Sophie Germain dated July 24,1826. Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 327, but with the year in error.
Draft of a letter from Sophie Germain lacking address and dated 18 July. Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 328. The draft’s contents reveal that the letter was intended for Cauchy. The date of Cauchy’s response establishes, in turn, the year it was written.
Cauchy’s memoirs in elasticity were not generally accessible until the period 1827–1829, when he published them in his Exercises de mathématiques. See his Oeuvres complètes, op. cit.,7, 8, and 9.
Bibliothèque Nationale, MS. Fr. 9118. Also in Stupuy, op. cit., p. 326, but with the year in error.
At the Academy’s session of 1 October, 1827, Poisson read a short note describing his research in progress on the vibration of solids. He asked the Academy’s permission to make known the basis of his analysis and some of its results even though he had not finished his work. At the conclusion of his reading, Cauchy announced that he too had been working for a long while on this subject - the equilibrium and movement of a solid body considered as a system of isolated molecules. At the end of the meeting Cauchy delivered a manuscript for review. (P. V., vol. 8, p. 603.) Poisson’s note was published in the Bull. des Sci. Math., (ed. Saigney), 9, (1828) 27–31. Cauchy’s molecular analysis was published as ‘Sur l’équilibre et le mouvement d’un système de points matériels sollicités par les forces d’attraction ou de repulsion mutuelle’. Exercices de mathématiques, année 1828. Oeuvres complètes, op. cit. 8, pp. 227–252. Navier’s complete memoir appears in Mém. Acad. Sci. 7, pp. 375–393.
Poisson, S. D.: 1828, ‘Mémoire sur l’équilibre et le mouvement des corps élastiques’. Annal. de chimie 37, pp. 337–355. His full treatise, carrying the same title, is in Mém. Acad. Sci. 8, pp. 357–570, 623-627.
Navier, C.: 1828, “Note relative à l’article intitulé: Mémoire sur l’équilibre’, Annal. de chimie 38, pp. 304–314.
Poisson responded with ‘Réponse à une note de M. Navier…’, Annal. de chimie 38, (1828), 435–440. Navier, in turn, submitted his ‘Remarques sur l’article de M. Poisson, inséré dans le cahier d’aôut, page 435, Annal. de chimie 39, (1828) 145–151. The polemic terminated with a letter from each participant to Arago, a convenient third party: ‘Lettre de M. Poisson à M. Arago’, Annal. de chimie 39. (1828)204–211 and ‘Lettre de M. Navier à M. Arago’, Annal. de chimie 40, (1829) 99–107.
Germain, S.: 1828, ‘Examen des principes qui peuvent conduire à la connaissance des lois de l’équilibre et du mouvement des solides élastiques’, Annal. de chimie 38, pp. 123–131.
Ibid.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1980 D. Reidel Publishing Company, Dordrecht, Holland
About this chapter
Cite this chapter
Bucciarelli, L.L., Dworsky, N. (1980). Emergence of a Theory. In: Sophie Germain. Studies in the History of Modern Science, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9051-7_9
Download citation
DOI: https://doi.org/10.1007/978-94-009-9051-7_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-277-1135-9
Online ISBN: 978-94-009-9051-7
eBook Packages: Springer Book Archive