Abstract
In this historical note we report an exchange of opinions between Giacinto Morera and Eugenio Beltrami concerning the representation of the general solution of the linear equilibrium equations. The expressions of the authors are interpreted and presented as stress equilibrium and strain compatibility in the tensor elastic field.
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Notes
Giacinto Morera (Novara, 18 July 1856–Turin, 8 February 1909), was an Italian engineer and mathematician. He is known for Morera’s theorem in the theory of functions of a complex variable and for his work in the theory of linear elasticity. He started his academic career in 1885 working at the University of Pavia as a professor in the then newly established “Scuola di Magistero”. In 1886 he became professor of rational mechanics at the University of Genova: he lived there for 15 years, serving also as dean and as rector. In 1901 he was called by the University of Turin to hold the chair of rational mechanics, left vacant by Vito Volterra. In 1908 he passed to the chair of “Meccanica Superiore” and was elected dean of the Faculty of Sciences.
Eugenio Beltrami (Cremona, 16 November 1835–Rome, 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. He began studying mathematics at University of Pavia in 1853; during this time he was taught and influenced by Francesco Brioschi. He had to discontinue his studies because of financial hardship and spent the next several years as a secretary working for the Lombardy–Venice railroad company. He was appointed to the University of Bologna as a professor in 1862, the year he published his first research paper. Throughout his life, Beltrami had various professorial jobs at the universities of Pisa, Rome and Pavia. From 1891 until the end of his life Beltrami lived in Rome. He became the president of the Accademia dei Lincei in 1898 and a senator of the Kingdom of Italy in 1899.
Following [7], we have for instance
$$\begin{aligned} \epsilon _{pil}\,\epsilon _{qjm}= \det \begin{bmatrix} \delta _{pq}&\delta _{pj}&\delta _{pm} \\ \delta _{iq}&\delta _{ij}&\delta _{im} \\ \delta _{lq}&\delta _{lj}&\delta _{lm} \end{bmatrix}. \end{aligned}$$Beltrami sets \((1+\nu )= C\), calling C “Clebsch’s constant” (Theorie der Elasticität fester Körper, B. G. Teubner 1862). There exists a french translation of Saint-Venant published in Paris in 1885.
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Acknowledgements
I thank particularly my colleague Alessandro Russo for his advices in writing this paper. I also thank the two anonymous reviewers for their valuable comments and suggestions.
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Landriani, G.S. A note about an exchange of opinions between Beltrami and Morera on elastic equilibrium equations. Meccanica 52, 2801–2806 (2017). https://doi.org/10.1007/s11012-016-0611-z
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DOI: https://doi.org/10.1007/s11012-016-0611-z