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The Theory of Elasticity in the 19th Century

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Strength of Materials and Theory of Elasticity in 19th Century Italy

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 52))

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Abstract

Until 1820 there was a limited knowledge about the elastic behavior of materials: one had an inadequate theory of bending, a wrong theory of torsion, the definition of Young’s modulus. Studies were made on one-dimensional elements such as beams and bars, and two-dimensional, such as thin plates (see for instance the work of Marie Sophie Germain). These activities started the studies on three-dimensional elastic solids that led to the theory of elasticity of three-dimensional continua becoming one of the most studied theories of mathematical physics in the 19th century. In a few years most of the unresolved problems on beams and plates were placed in the archives. In this chapter we report briefly a summary on three-dimensional solids, focusing on the theory of constitutive relationships, which is the part of the theory of elasticity of greatest physical content and which has been the object of major debate. A comparison of studies in Italy and those in the rest of Europe is referenced.

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Notes

  1. 1.

    vol. 4, pp. 349, 350.

  2. 2.

    pp. 344–345. Our translation.

  3. 3.

    Addition, pp. 79–81.

  4. 4.

    pp. 375–376. Our translation.

  5. 5.

    p. 384.

  6. 6.

    p. 384.

  7. 7.

    The difficulty of replacing summations with integrals has been the subject of many comments of French scholars, especially Poisson and Cauchy.

  8. 8.

    pp. CLV, CLIX. The memoir of Cauchy appeared first with the title Mémoire sur l’équilibre et le mouvement d’un système de points materiels sollecités par forces d’attraction ou de répulsion mutuelle [30]. That of Poisson appeared with the title Note sur les vibrations des corps sonores [126].

  9. 9.

    pp. 368–369. Our translation.

  10. 10.

    p. 29. Our translation. Stress was indicated by French scientists by pressure or tension.

  11. 11.

    pp. 250–251.

  12. 12.

    p. 403.

  13. 13.

    pp. 227–252.

  14. 14.

    Actually Cauchy introduced various slightly different definitions of stress. In a memoir of 1845 [34] he adopted the definition considered also by Saint Venant and Jean-Marie Constant Duhamel according to which the “stress (la pression) on a very small area (\(\upomega \) is defined) as the resultant of the actions of all the molecules located on the one side over all the molecules located on the other side whose directions cross this element” [141], p. 24.

  15. 15.

    p. 257.

  16. 16.

    p. 257, Eq. (1.13).

  17. 17.

    pp. 60–81.

  18. 18.

    p. 260, Eq. (1.18).

  19. 19.

    pp. 162–173.

  20. 20.

    p. 9.

  21. 21.

    pp. 83–85.

  22. 22.

    See the results found by Guillaume Wertheim (1815–1861) [158, pp. 581–610]. The greater the accuracy and reliability of the experimental results the more the theoretical predictions of Cauchy and Poisson were disclaimed, though it was not clear why [80, pp. 481–503].

  23. 23.

    p. 69. Our translation.

  24. 24.

    p. 85. Our translation.

  25. 25.

    pp. 7–8. Our translation.

  26. 26.

    s. 2, vol XI, pp. 11–27; 51–74; 134–172.

  27. 27.

    Appendix V, p. 689.

  28. 28.

    A detailed reconstruction of Cauchy’s topics is shown in [116], Appendix V, pp. 691–706.

  29. 29.

    pp. 51–52. Our translation.

  30. 30.

    pp. 36–37.

  31. 31.

    p. 38. Our translation.

  32. 32.

    pp. 542–543. Our translation.

  33. 33.

    p. 759. Our translation.

  34. 34.

    p. 556. Our translation.

  35. 35.

    pp. 556–560.

  36. 36.

    p. 582. Our translation.

  37. 37.

    p. 583. Our translation.

  38. 38.

    In the order: pp. 353–430; 297–350; 242–254.

  39. 39.

    For a discussion of the positivistic conceptions of French science in the first half of the 19th century, see [124].

  40. 40.

    It seems that on September 30th 1822, Cauchy notified the Académie of his researches neither delivering a public reading, nor depositing a manuscript; see [3] p. 97. In [154] it is stated that Cauchy, as a matter of fact, presented his memoir.

  41. 41.

    Cauchy used tension or pressure for traction and compression respectively.

  42. 42.

    p. 300. Our translation.

  43. 43.

    p. 144.

  44. 44.

    The term tensor does not belong to Cauchy, but to Hamilton [76] and Voigt [156]. Similarly Cauchy did not establish the formalized rules of tensor calculus, that were specified only at the end of the 19th century by Ricci-Curbastro; see in particular [89, pp. 125–201].

  45. 45.

    It should be noted that, in all the above mentioned works, Cauchy made extensive use of infinitesimals, whereas he had pursued his research in mathematical analysis with the precise goal of eliminating the infinitesimals. This attitude is similar to that held by Lagrange who while in the Théorie des fonctions analytiques of 1797 developed a way to avoid the use of infinitesimals, in the Méchanique analitique of 1788 extensively applied the infinitesimals, justifying their use for the sake of simplicity [17].

  46. 46.

    Consequently the quadrilateral is a parallelogram whose contiguous sides, (1, 2), (1, 3), will be [...]:

    $$\begin{aligned}(1,2)=dx\left( 1+\frac{\partial \updelta x}{\partial x}\right) ;\;(1,3)=dy\left( 1+\frac{\partial \updelta y}{\partial y}\right) , \end{aligned}$$

    [...] with respect to the angle included by this two sides one will find [...]:

    $$\begin{aligned}\cos \upalpha =\frac{\partial \updelta x}{\partial y}+\frac{\partial \updelta y}{\partial x}. \end{aligned}$$

    [85], pp. 208–209. Our translation. See also [61], pp. 288–292; 332–334.

  47. 47.

    p. 209.

  48. 48.

    p. 215.

  49. 49.

    p. 216. Actually the first three equations were written by Cauchy in a slightly different way, though equivalent to that referred above.

  50. 50.

    p. 218, Eq. 76.

  51. 51.

    Indicated by Lamé with the symbols \(N_i, T_i, \; i=1,2,3\), \(N_i\) being the normal component and \(T_i \) the tangential one, to the face on which the force acts.

  52. 52.

    p. 33.

  53. 53.

    p. 50.

  54. 54.

    p. 51.

  55. 55.

    p. 6. Our translation.

  56. 56.

    p. 249.

  57. 57.

    p. 245.

  58. 58.

    We already said at the end of the previous section that Green introduced the six components of the infinitesimal strain before Saint Venant. He indicated with \(s _ 1, s_2, s_3 \) the longitudinal strains, which are equal to the percentage change of the edge lengths \(dx, dy, dz \) of an elementary parallelepiped and with \( \upalpha , \upbeta , \upgamma \) the angular distortions, equivalent to the variation of the angles between edges initially orthogonal \(dy\) and \(dz\), \(dx\) and \(dz\), \(dx\) and \(dy\).

  59. 59.

    p. 249.

  60. 60.

    p. 255.

  61. 61.

    p. 41. Our translation.

  62. 62.

    p. 708. Our translation.

  63. 63.

    p. 64.

  64. 64.

    From an unpublished manuscript quoted in [58], p. 331.

  65. 65.

    p. 747. Our translation.

  66. 66.

    p. VII. Our translation.

  67. 67.

    p. VIII. Our translation.

  68. 68.

    p. 196. Our translation.

  69. 69.

    p. 97. Our translation. Cauchy-Poisson’s relations are those relations that reduce from 36 to 15 the independent elastic constants of the more general elastic relationship; one in the case of isotropic bodies.

  70. 70.

    p. 288. Our translation.

  71. 71.

    p. 288. Our translation.

  72. 72.

    p. 289. Our translation.

  73. 73.

    p. 293. Our translation.

  74. 74.

    See Fig. 1.2a of paragraph 1.1.1, where \(e\) are the molecules in A\('\) and \(i\) the molecules in B.

  75. 75.

    p. 293. Our translation.

  76. 76.

    p. 293.

  77. 77.

    p. 596. Our translation.

  78. 78.

    p. 597. Our translation.

  79. 79.

    p. 597. Our translation.

  80. 80.

    For a more in depth study on Voigt’s work see [20].

  81. 81.

    An interesting comment of Born’s analysis can be found in [67], where also considerations on modern studies are referred to.

  82. 82.

    pp. 125–129.

  83. 83.

    The last reissue is entitled Principles of mechanics and dynamics, Dover, New York, 2003. The work was begun in 1861; programmed in multiple volumes, for commitments of the two authors, it saw the light of day with only the first volume.

  84. 84.

    vol. 1, p. 136.

  85. 85.

    vol. 1, p. V.

  86. 86.

    In structural mechanics those systems of bodies that contain more constraint reactions than equilibrium equations are called statically indeterminate. The difference between the number of constraint reactions and equilibrium equations is called degree of hyperstaticity. The degree of hyperstaticity can be defined also in dual mode as the difference between the elementary constraints and degrees of freedom of the system of bodies. Because the constraint reactions cannot be determined with the equations of statics, using an engineering terminology, we say that the static problem is not determinate.

  87. 87.

    p. IX. Our translation.

  88. 88.

    For comment on this issue see [40].

  89. 89.

    Here Möbius studied plane and spatial trusses, found the minimum ratios between nodes and bars for a trusses to be statically determined (for \(n \) nodes one needs \( 2n-3 \) bars in the plane and \(3n-6 \) in the space) and discovered that the minimum requirement is not sufficient for the equilibrium if the static matrix has zero determinant. Möbius’ work, at least initially, was not noticed by engineers, thus deserving the considerations for the contribution of the precursors (see next sections). The results of Möbius were found again (likely) independently by Christian Otto Mohr in 1874.

  90. 90.

    Ritter’s method was developed in embryonic form by Culmann in two articles, of 1851 [52] and of 1852 [53], which showed the state of art of iron and wood bridges in Great Britain and America. Here Culmann used equilibrium equations of forces and moments for the determination of the stresses in bars. In particular, the equations of equilibrium were related to sections of the truss. Culmann results were taken again by August Ritter (1826–1908) in his textbook of 1863 Elementare Theorie der Dach und Brüken-Constructionen [136]. Ritter also known in astrophysics (in his honor the Ritter’s crater on the lunar surface), was formed in Hannover and Göttingen; from 1856 he taught mechanics in Hannover, then in 1870 at the Technical college of Aachen. Although in Ritter’s textbook there were no evident references to Culmann (about the method of sections he mentioned only one of his previous work in the journal of the architects and engineers of Great Britain), it is likely that his work, set on roofs of buildings and bridges, was in any way influenced by the articles of Culmann. The method of ‘Ritter sections’, can also be associated to another Ritter, Karl Wilhelm Ritter (1847–1906), who graduated in Zurich in 1868 and, after some professional activities in Hungary, was assistant of Culmann in Zurich from 1870 to 1873. Karl Wilhelm Ritter taught at the Polytechnic institute of Riga until 1882, when, following the death of the master, he was called to the chair of Zurich. He is the author of a key text on trusses [137] where the method of sections is used.

  91. 91.

    p. 290. Our translation.

  92. 92.

    p. 290. Our translation.

  93. 93.

    A similar approach is used to determine the axial stresses in the biaxial bending of a Saint Venant’s cylinder.

  94. 94.

    p. 291. Our translation.

  95. 95.

    vol II/1, p. 411.

  96. 96.

    p. cviii.

  97. 97.

    vol. 1, p. 241. Our translation.

  98. 98.

    vol. 1, p. 235.

  99. 99.

    p. CCXII.

  100. 100.

    p. 953. Our translation.

  101. 101.

    pp. 405–406.

  102. 102.

    p. 1077.

  103. 103.

    p. 1078. Our translation.

  104. 104.

    The force of elasticity is the axial stiffness, that is the product of the longitudinal elasticity modulus by the area of the cross section of the bars, which multiplied by the axial strain gives the force in the bar.

  105. 105.

    vol.1, pp. 346–347. Our translation.

  106. 106.

    vol. 1, p. 347. Our translation.

  107. 107.

    p. 410. Our translation.

  108. 108.

    p. 411. Our translation.

  109. 109.

    vol 2, pp. 402–404.

  110. 110.

    Clebsch’s text had widespread diffusion only in the late nineteenth century, thanks to the translation into French by Saint Venant [42].

  111. 111.

    The Eq. (1.4) are the three equations of motion of a material point \(m\) subjected to external forces and constraints \(L=L'=L^{\prime \prime }=0\); we refer here to one of them as an example: \(m \displaystyle \frac{d^2x}{dt^2}=mX+\lambda ^{\prime } \displaystyle \frac{dL}{dx}+\lambda \displaystyle \frac{dL^{\prime }}{dx}+\lambda ^{\prime \prime } \displaystyle \frac{dL^{\prime \prime }}{dx}+ \text {etc.}\)

  112. 112.

    Previously Poisson had studied the motion under fixed constraints where \(L, L^{\prime }, L^{\prime \prime }, L^{\prime \prime \prime }\) were the constraint equations and \(\lambda , \lambda ^{\prime }, \lambda ^{\prime \prime }, \lambda ^{\prime \prime \prime }\) the constraint reactions.

  113. 113.

    vol 2, pp. 402–404. Our translation.

  114. 114.

    p. 24; p. 69. Actually Lagrange used the symbol \(V\) in the Théorie de la libration de la Lune and the symbols \(\Phi \) and \(\Pi \) respectively in the first and subsequent editions of the Mécanique analytique.

  115. 115.

    p. 386.

  116. 116.

    pp. 1–82.

  117. 117.

    p. 784.

  118. 118.

    p. 57.

  119. 119.

    p. 1197.

  120. 120.

    pp. 79–83. Our translation.

  121. 121.

    p. 217.

  122. 122.

    pp. CXCVII–CXCVIII. Our translation.

  123. 123.

    In the history of science there are people who are ahead of one’s time, anticipating theories. Some are precursors only in appearance and look like such to us because we are strangers to the cultural climate of the time. When the precursors are real, their lack of success depends on contingent reasons, such as poor prestige enjoyed and the publication of pioneering studies in journals not known to those who might appreciate them. For those who understand the history of science in a cumulative sense, precursors disturb the linear path that one wants to follow. For others, the study of the precursors is of interest, although not central: the understanding of their ideas is helpful in understanding the cultural climate of the time. Maxwell and Cotterill should be considered from this second point of view. Together with precursors one must also consider successors, those who come to a result after its spread among the specialists. This stems from the conditions of cultural isolation; also the study of successors is interesting in understanding how scientific ideas grow in a given cultural climate. As part of the history of structural mechanics, among the successors we believe to deserve being mentioned are Wilhelm Fränkel [69] who regained results of Menabrea and Castigliano; and Friedrich Engesser [62] who regained Crotti’s results (see Chap. 4) and introduced the term complementary work [51]. Engesser defined complementary work as the difference between actual and virtual work.

  124. 124.

    p. 296.

  125. 125.

    p. 297.

  126. 126.

    p. 294.

  127. 127.

    p. 456.

  128. 128.

    p. 296.

  129. 129.

    p. 296.

  130. 130.

    p. 298.

  131. 131.

    pp. 81–83.

  132. 132.

    Chapter 10.

  133. 133.

    He was the lucky author of The mechanical principles of engineering and architecture [110]; see [153] on pp. 212–214.

  134. 134.

    The principle of least resistance by Moseley is: “If any number of pressures are equilibrated, some of which are resistances, then each of these resistances is a minimum, subjected to the conditions imposed by the equilibrium of the whole” [108, 109], p. 178, [46], p. 299.

  135. 135.

    p. 300.

  136. 136.

    p. 301.

  137. 137.

    p. 303.

  138. 138.

    p. 388–389.

  139. 139.

    p. 305.

  140. 140.

    pp. 301–302.

  141. 141.

    p. 383.

  142. 142.

    pp. 1060–1061. Our translation.

  143. 143.

    vol. 4, pp. 210–211. Our translation.

  144. 144.

    vol. 4, p. 211. Our translation.

  145. 145.

    vol. 4, p. 213.

  146. 146.

    col. 225.

  147. 147.

    The difference between external and internal bars concerns the contribution of the load \(P=1\), which for the internal bars gives the moment \(a \cdot 1=a\), constant with respect to any pole.

  148. 148.

    col. 226. Our translation.

  149. 149.

    col. 229–230. Our translation.

  150. 150.

    We assigned a positive sign to the horizontal thrust \(H\); the shortening \(\Delta s\) is negative, so \(-H \cdot \Delta s\) is a positive magnitude (original note by Mohr).

  151. 151.

    The quantity \(u \cdot H \cdot \Delta l\) is always positive, since \( \Delta l\) is a lengthening or a shortening whenever \(u \cdot H \) is a tensile or compressive stress. The quantities \( \Delta l\) and \(u \cdot H\) have so in this paper always the same sign (original note by Mohr).

  152. 152.

    col. 231–232. Our translation.

  153. 153.

    col. 512. Our translation.

  154. 154.

    col. 517–518.

  155. 155.

    col. 517–518. Our translation.

  156. 156.

    col 17–38. 17–38. The article has a different typeface with respect to the preceding works with the same title: there is no longer the character ‘Fraktur’, a gothic font typical of the more traditional German literature.

  157. 157.

    cols. 20–22.

  158. 158.

    cols. 22–29.

  159. 159.

    Italian translation of 1927, p. 534. Our translation.

  160. 160.

    A review of the situation of Italian mathematics in the early 19th century, along with an extensive bibliography, can be found in [13]. At the beginning of the book, p. 23, a depressing commentary of 1794 by Pietro Paoli, professor at the study of Pisa is referred to: “Among all those who in Italy dedicate to the study of mathematics, if we exclude some genius, [...] there are a few others that come to mediocrity [...] [most people] at the first reading of the books by Euler, D’Alembert, and Lagrange, get into insurmountable difficulties” [123], vol. 1, p. V.

  161. 161.

    Lorenzo Mascheroni (Bergamo 1750–Paris 1800). Mathematician, his most important contributions concerned mathematical analysis, including studies related to integral calculus and logarithms, structural mechanics with his original studies on the breaking of arches and geometry, with a demonstration that the problems solvable with ruler and compass can be solved only with a compass.

  162. 162.

    Vittorio Fossombroni (Arezzo 1754–Florence 1844). Mathematician, engineer, economist and politician. Important is his contribution to the development of the principle of virtual work.

  163. 163.

    Girolamo Saladini (Lucca 1740–1813). Mathematician, pupil of Vincenzo Riccati, an early member of the Società dei XL (see below), an often quoted mathematician.

  164. 164.

    Vincenzo Angiulli (Ascoli Satriano 1747–1819). Mathematician and politician. Important work was his Discorso intorno agli equilibri of 1770, where he developed and clarified the contribution of Vincenzo Riccati to the principle of virtual work [21].

  165. 165.

    Michele Araldi (Modena 1740–Milano 1813). Physicist and mathematician, historian of mathematics and physics of his time, he wrote the histories of contemporary mathematics in the prefaces of the Memorie dell’Istituto Italiano. He was among the first members of the Regio istituto Lombardo.

  166. 166.

    Antonio Maria Lorgna, also known as Anton Maria Lorgna or Mario Lorgna (Ceredigion 1735–Verona 1796). Mathematician, astronomer and engineer. In 1782 he promoted the foundation of the Società italiana delle scienze, which edited the Memorie di matematica e fisica. Being among the first founders, forty in number, the society was also called Società dei XL and still operates with this name.

  167. 167.

    Paolo Delanges (1750 c.a. –1810). Mathematician, student of Vincenzo Riccati. In 1803 he was appointed a member of the Istituto Nazionale della Repubblica Italiana, based in Bologna. He was one of the first members of the Società dei XL.

  168. 168.

    p. CCXCIII.

  169. 169.

    vol 1, p. 419.

  170. 170.

    vol. 1, p. 460.

  171. 171.

    vol 1, p. 656.

  172. 172.

    p. 224. Our translation.

  173. 173.

    p. 181.

  174. 174.

    The reference to Castigliano does not seem quite right. He, in fact, despite the energetic background, due to the great attention given to elastic energy, was significantly deployed on Saint Venant’s molecular positions. In his La théorie de l’équilibre des systèmes élastiques et ses applications Castigliano criticized the failure criteria based on maximum stress or maximum strain, but did not suggest an alternative criterion [24].

  175. 175.

    p. 189. Our translation.

  176. 176.

    The ranking of the graduation thesis was done on the average of the votes of the individual exams (11 exams in 2  years) which was compounded by the vote of the thesis dissertation. In the case of Cerruti the average was 318/330 for the 11 exams and 348/360 the final average. The thesis was assessed 30/30. Castigliano gained an average of 313/330 and the vote of the thesis 30/30. The final vote was 343/360 (Private communication by Margherita Bongiovanni).

  177. 177.

    p. 1056. Our translation.

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Capecchi, D., Ruta, G. (2015). The Theory of Elasticity in the 19th Century. In: Strength of Materials and Theory of Elasticity in 19th Century Italy. Advanced Structured Materials, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-05524-4_1

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