Skip to main content
SpringerLink
Log in

Antonio Signorini and the proto-history of the non-linear theory of elasticity

  • Published:
Archive for History of Exact Sciences Aims and scope Submit manuscript

Abstract

Antonio Signorini’s contribution to the constitutive theory of non-linear elasticity is reconstructed and analyzed. Some uninformed opinions suggesting he had a minor role, lacking of significant results, are discussed and refuted. It is shown that Signorini should be rightly credited for being among the first scholars aware of the central problem of non-linear elasticity: the determination of the general form of the elastic potential.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. Tesi di Laurea.

  2. This series of congresses are today known as “ICTAM: International Conference of Theoretical and Applied Mechanics”.

  3. This quote is from the material added in the preface to the third edition of Truesdell and Noll (1965), published in 2004.

  4. “Fatta più col cuore che col cervello”.

  5. Associazione Italiana di Meccanica Teorica ed Applicata.

  6. “L’uso della lingua italiana, poco conosciuta fuori d’Italia, i contatti poco frequenti tra gli Studiosi e le difficoltá presentate dal difficile algoritmo delle omografie, strumento preferito da Signorini, hanno fatto si che la Sua opera fosse poco conosciuta all’estero e, a volte, nel nostro stesso Paese.”

  7. Omografia (in Italian).

  8. A first edition of this text was published in French in Burali-Forti and Marcolongo (1912).

  9. Omografia vettoriale.

  10. “Una trattazione assoluta di tutte le questioni fisico-meccaniche-geometriche, indipendentemente da qualsiasi sistema di coordinate.”

  11. “Venti anni di lotte, di studi e di un lavoro condotto con fede e con costanza.”

  12. On the basis of Picone’s and Fichera’s recollections, Signorini in the last months of his life, already confined at home for health reasons, often and apprehensively asked for progress on the solution of this problem.

  13. Istituto Nazionale di Alta Matematica. This congress had very little international scope due to the world war situation. In fact, only sixteen mathematicians from outside Italy, and only from ‘allied’ or neutral countries, were present: no English or French mathematicians at all.

  14. “...prima di ogni calcolo numerico si ha da affrontare un problema estremamente difficile di vera Fisica matematica: la scelta dell’espressione completa del potenziale elastico.”

  15. “Il problema che ben si può dire il problema centrale dell’intera teoria non linearizzata, è il problema della scelta dell’espressione completa del potenziale elastico.”

  16. Today, this center is known as the Tun Abdul Razak Research Centre.

  17. For Hadamard, a mathematical model of a physical phenomenon is well posed if: (i) the solution exists; (ii) the solution is unique, and (iii) the solution’s behavior depends with continuity on the initial conditions.

  18. In Truesdell and Noll (1965) we read: “The radically simpler nature of the incompressible material in finite deformations was seen by Rivlin and exploited in a sequence of remarkable papers in 1948–1954.”

  19. The full determination of such family of strain-energy densities can be found in Mangan et al. (2016).

  20. See, for example, the review article (Savin and Koifman 1970) published in 1970.

  21. Georg Hamel (1877–1953).

  22. “Ho fatto vedere che il problema centrale della teoria delle trasformazioni termoelastiche finite, in sostanza si riduce alla scelta del potenziale isotermo \(W_{\tau }\) per ogni valore di \(\tau \).”

  23. “Questa mia presunzione venne avvalorata, fin dal 1955, da un fatto almeno per me inaspettato: per sistemi incomprimibili l’ipotesi caratteristica dell’Elasticitá di secondo grado impone al potenziale isotermo una forma che – ove si annulli uno dei tre parametri in essa disponibili – è proprio la forma proposta da Mooney nel 1950 e poi sottoposta con successo anche a svariati controlli sperimentali”.

  24. A female student of Vito Volterra for whom, at the moment, there are no relevant historical notices.

  25. Ariano’s scientific production would certainly deserve an appropriate discussion, out of the scope of the present manuscript.

  26. It is clear that Brillouin obtained his result independently of Crudeli. They were completely different scientists with different interests. This topic was only a tiny intersection of their scientific production.

  27. GAMM’s conference in 1955, whose English translation by Truesdell himself can be found in Truesdell (1965a).

  28. Truesdell makes use of the letter \(\Sigma \) rather than W.

  29. Here we are using the term robust in the usual sense of the philosophy of science (Soler et al. 2012).

  30. \(\Sigma \) is the symbol for the strain-energy density.

  31. We notice another strange situation about papers Truesdell (1952b, 1952c). Let us denote them, conventionally, as the first and the second one. In the second one (Truesdell 1952c) (received by the journal on July 16, 1951) Truesdell cites the first paper saying “The material treated here and many other related topics are summarized in Truesdell (1952b)”. It is clear that the first paper is not a summary of the second one because since 100 pages are much more than the 20 pages of the second one, which is cited in Truesdell (1952b) as “forthcoming”.

  32. See http://archiviovasari.beniculturali.it/index.php/il-rapporto-con-michelangelo/.

  33. Con questa Memoria l’A. continua l’esposizione sistematica delle sue ricerche nell’ambito della Elasticitá non linearizzata, per dare completo sviluppo a quanto più o meno sinteticamente giá si trova esposto in conferenze tenute a Stoccolma [\(3^\text {o}\) Congr. int. di Mecc. appl., 1930], a Palermo [XXIV Riunione della S. L P. S., 1935], a Bologna [\(2^\text {o}\) Congr. della U. M. I.,1940], a Roma [Convegno mat. del 1942] e a Londra [\(7^\text {o}\) Congr. int. di Mecc. appl., 1948].

  34. It is well known that Biot’s work was somewhat “censored” in Truesdell and Noll (1965), where the theory of small on large deformations has been considered in details but only one paper by Biot on waves is cited, with no comments at all.

  35. Questa seconda memoria non richiede una speciale introduzione, perché i principali suoi risultati giá si trovano esposti e commentati nelle pubblicazioni citate nel Sunto.

  36. “Sulla deduzione del potenziale elastico dall’esperienza”

  37. “Non ragioniam di lor, ma guarda e passa” Dante, Inf. III, 51.

  38. “...poi richiamare l’attenzione su certi fatti non banali che possono presentarsi nella teoria non linearizzata: “non banali” nel senso che completamente sfuggono alla teoria classica.”

  39. Nikoloz (Niko) Muskhelishvili 1891–1976.

  40. “Nel settembre scorso, durante il congresso internazionale, tenutosi a Tihilis, in Georgia dalla Societá Internazionale di Meccanica Applicata, Antonio Signorini fu ricordato con commosse parole dal Presidente del Congresso, il grande matematico russo Muskhelishvili”.

References

  • Almansi, E. 1911. Sulle deformazioni finite dei solidi elastici isotropi. Rendiconti della Reale Accademia dei Lincei: Classe di Scienze Fisiche, Matematiche e Naturali 20: 705–714.

  • Ariano, R. 1924. Deformazioni finite di sistemi continui. Rendiconti del Circolo Matematico di Palermo 48 (1): 97–120.

    Article  MathSciNet  Google Scholar 

  • Ariano, R. 1925. Deformazioni finite di sistemi continui. Annali di Matematica Pura ed Applicata 2 (1): 217–261.

    Article  MathSciNet  Google Scholar 

  • Armanni, G. 1915. Sulle deformazioni finite dei soldi elastici isotropi. Il Nuovo Cimento 9 (1): 427–447.

    Article  Google Scholar 

  • Ball, J.M. 1976. Convexity conditions and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis 63 (4): 337–403.

    Article  MathSciNet  Google Scholar 

  • Ball, J.M., and R.D. James. 2002. The scientific life and influence of Clifford Ambrose Truesdell III. Archive for Rational Mechanics and Analysis 161 (1): 1–26.

    Article  MathSciNet  Google Scholar 

  • Bordoni, P.G. 1953. Sopra le trasformazioni termoelastiche finite di certi solidi omogenei ed isotropi. Rendiconti di Matematica e delle sue Applicazioni V 12: 3–4.

    MathSciNet  Google Scholar 

  • Brillouin, L. 1925. Sur les tensions de radiation. Annales de Physique 4: 528–586.

    Article  Google Scholar 

  • Burali-Forti, C., and R. Marcolongo. 1912. Analyse Vectorielle Générale - Transformations Linéaires. Pavia: Mattei & C. Éditeurs.

    Google Scholar 

  • Burali-Forti, C., and R. Marcolongo. 1929. Analisi Vettoriale Generale e Applicazioni - Trasformazioni Lineari, vol. 1. Bologna: Zanichelli.

    Google Scholar 

  • Capriz, G., and P. Podio-Guidugli. 1974. On Signorini’s perturbation method in finite elasticity. Archive for Rational Mechanics and Analysis 57: 1–30.

    Article  MathSciNet  Google Scholar 

  • Cesàro, E. 1894. Introduzione Alla Teoria Matematica della Elasticità, vol. 7. Torino: Fratelli Bocca.

    Google Scholar 

  • Cosserat, E., and F. Cosserat. 1896. Sur la théorie de l’élasticité. Premier mémoire. Annales de la Faculté des Sciences de Toulouse: Mathématiques 10 (3–4): I1–I116.

    MathSciNet  Google Scholar 

  • Crudeli, U. 1917. Di una particolare teoria delle deformazioni elastiche finite. Il Nuovo Cimento 14 (1): 125–130.

    Article  Google Scholar 

  • Destrade, M., J. Murphy, and G. Saccomandi. 2019. Rivlin’s legacy in continuum mechanics and applied mathematics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377 (2144): 20190090.

    Article  MathSciNet  Google Scholar 

  • Destrade, M., G. Saccomandi, and I. Sgura. 2017. Methodical fitting for mathematical models of rubber-like materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473 (2198): 20160811.

    Article  Google Scholar 

  • Fichera, G. 1972. Sulla propagazione delle onde in un mezzo elastico. Continuum Mechanics and Related Problems of Analysis: 567–574.

  • Fichera, G. 1973. Boundary Value Problems of Elasticity with Unilateral Constraints, Linear Theories of Elasticity and Thermoelasticity, 391–424. Heidelberg: Springer.

    Book  Google Scholar 

  • Green, A.E., R.S. Rivlin, and R.T. Shield. 1952. General theory of small elastic deformations superposed on finite elastic deformations. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 211 (1104): 128–154.

    MathSciNet  Google Scholar 

  • Green, A.E., and R.T. Shield. 1951. Finite extension and torsion of cylinders. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 244 (876): 47–86.

    MathSciNet  Google Scholar 

  • Grioli, G. 1962. Mathematical Theory of Elastic Equilibrium (Recent Results), Ergebnisse der Angewandten Mathematik, vol. 7. Berlin: Springer.

    Book  Google Scholar 

  • Grioli, G. 1991. L’opera di Antonio Signorini nella Fisica Matematica. In Opere Scelte, ed. A. Signorini, XI–XXIV, Firenze: Edizioni Cremonese (distribuito dall’Unione Matematica Italiana).

  • Love, A.E.H. 1892. A Treatise on the Mathematical Theory of Elasticity. Cambridge: Cambridge University Press.

    Google Scholar 

  • Lurie, A.I. 2010. Theory of Elasticity. Berlin: Springer.

    Google Scholar 

  • Lurie, A.I. 2012. Non-linear Theory of Elasticity. Amsterdam: North-Holland.

    Google Scholar 

  • Manacorda, T. 1956. Sul potenziale isotermo nella più generale elasticità di secondo grado per solidi incomprimibili. Annali di Matematica Pura ed Applicata 41 (1): 77–86.

    Article  MathSciNet  Google Scholar 

  • Mangan, R., M. Destrade, and G. Saccomandi. 2016. Strain energy function for isotropic non-linear elastic incompressible solids with linear finite strain response in shear and torsion. Extreme Mechanics Letters 9: 204–206.

    Article  Google Scholar 

  • Mooney, M. 1940. A theory of large elastic deformation. Journal of Applied Physics 11: 582–592.

    Article  Google Scholar 

  • Picone, M. 1965. Commemorazione di Antonio Signorini, Atti e Memorie della Accademia Petrarca di Lettere (1958–1964) Nuova Serie, XXXVII, 378–402. Arezzo: Arti e Scienze.

    Google Scholar 

  • Pöschl, T. 1930. Dritter internationaler kongreß für technische mechanik vom 24. bis 29. August in Stockholm 18: 937–939.

    Google Scholar 

  • Rivlin, R.S. 1948a. Large elastic deformations of isotropic materials, I. Fundamental concepts. Philosophical Transactions of the Royal Society of London. Series A 240: 459–490.

    Article  MathSciNet  Google Scholar 

  • Rivlin, R.S. 1948b. Large elastic deformations of isotropic materials, II. Some uniqueness theorems for pure, homogeneous deformation. Philosophical Transactions of the Royal Society of London. Series A 240: 491–508.

    Article  MathSciNet  Google Scholar 

  • Rivlin, R.S. 1948c. Large elastic deformations of isotropic materials, III. Some simple problems in cylindrical polar co-ordinates. Philosophical Transactions of the Royal Society of London. Series A 240: 509–525.

    Article  MathSciNet  Google Scholar 

  • Rivlin, R.S. 1948d. Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philosophical Transactions of the Royal Society of London. Series A 241: 379–397.

    Article  MathSciNet  Google Scholar 

  • Rivlin, R.S., and D.W. Saunders. 1951. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philosophical Transactions of the Royal Society of London. Series A 243: 251–288.

    Article  Google Scholar 

  • Savin, G.N., and Y.I. Koifman. 1970. The general nonlinear theory of elasticity. Soviet Applied Mechanics 6: 1275–1295.

    Article  Google Scholar 

  • Seth, B.R. 1935. Finite strain in elastic problems. Philosophical Transactions of the Royal Society of London. Series A 234: 231–264.

    Article  Google Scholar 

  • Signorini, A. 1943. Trasformazioni termoelastiche finite (Memoria 1a̱). Annali di Matematica Pura ed Applicata 22 (4): 33–143.

    Article  MathSciNet  Google Scholar 

  • Signorini, A. 1948. On finite deformations of an elastic solid. In Proc. Seventh Internat. Congress Appl. Mech. V vol. 4, p. 237–247.

  • Signorini, A. 1949. Trasformazioni termoleastiche finite (Memoria 2a̱). Annali di Matematica Pura ed Applicata 30 (4): 1–72.

    Article  MathSciNet  Google Scholar 

  • Signorini, A. 1954. Necrologio di von Mises. Bollettino dell’Unione Matematica Italiana. Serie 3 9 (2): 228–229.

    Google Scholar 

  • Signorini, A. 1955. Trasformazioni termoelastiche finite (Memoria 3a̱). Annali di Matematica Pura ed Applicata 39 (4): 147–201.

    Article  Google Scholar 

  • Signorini, A. 1959a. Questioni di elasticità non linearizzata e semilinearizzata. Rendiconti di Matematica e Delle Sue Applicazioni 18: 95–139.

    MathSciNet  Google Scholar 

  • Signorini, A. 1959b. Résultats simples de la théorie non linéarisée de l’élasticité. Annales de la Faculté des Sciences de Toulouse: Mathématiques 23: 5–21.

    MathSciNet  Google Scholar 

  • Signorini, A. 1960. Trasformazioni termoelastiche finite (Memoria 4a̱). Annali di Matematica Pura ed Applicata 51 (4): 329–372.

    Article  MathSciNet  Google Scholar 

  • Signorini, A. 1961. Risultati semplici della teoria non linearizzata dell’elasticità. Rendiconti del Seminario Matematico e Fisico di Milano 31: 35–53.

    Article  Google Scholar 

  • Šilhavý, M. 2019. Consequences of the Coleman-Noll inequality for isotropic materials. ZAMM-Journal of Applied Mathematics and Mechanics (Zeitschrift für Angewandte Mathematik und Mechanik) 99 (6): e201800324.

    Article  MathSciNet  Google Scholar 

  • Soler, L., E. Trizio, T. Nickles, and W.E. Wimsatt. 2012. Characterizing the Robustness of Science: After the Practice Turn in Philosophy of Science, vol. 292. Berlin: Springer.

    Book  Google Scholar 

  • Tanner, R.I., and E. Tanner. 2003. Heinrich Hencky: a rheological pioneer. Rheologica Acta 42: 93–101.

    Article  Google Scholar 

  • Taylor, G.I. 1950. Seventh international congress for applied mechanics, 1948. Nature 165: 258–260.

    Article  Google Scholar 

  • Tolotti, C. 1943. Deformazioni elastiche finite: onde ordinarie di discontinuità e caso tipico di solidi elastici isotropi. Rendiconti di Matematica e delle Due Applicazioni 5: 33–59.

    MathSciNet  Google Scholar 

  • Treloar, L.R.G. 1943. The elasticity of a network of long-chain molecules. i. Transactions of the Faraday Society 39: 36–41.

    Article  Google Scholar 

  • Truesdell, C., and R. Toupin. 1960. The classical field theories. In Principles of Classical Mechanics and Field Theory / Prinzipien der Klassischen Mechanik und Feldtheorie, ed. S. Flügge, 226–858. Berlin: Springer.

    Chapter  Google Scholar 

  • Truesdell, C.A. 1952a. F. D. Murnaghan’s Finite deformation of an elastic solid. Bulletin of the American Mathematical Society 58: 577–579.

    Article  MathSciNet  Google Scholar 

  • Truesdell, C.A. 1952b. The mechanical foundations of elasticity and fluid dynamics. Journal of Rational Mechanics and Analysis 1: 125–300.

    MathSciNet  Google Scholar 

  • Truesdell, C.A. 1952c. A program of physical research in classical mechanics. Zeitschrift für Angewandte Mathematik und Physik ZAMP 3: 79–95.

    Article  MathSciNet  Google Scholar 

  • Truesdell, C.A. 1956. Das ungelöste hauptproblem der endlichen elastizitätstheorie. ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 36: 97–103.

    Article  Google Scholar 

  • Truesdell, C.A. 1965a. Continuum Mechanics: Foundations of Elasticity Theory, International Science Review Series, vol. III. New York: Gordon and Breach.

    Google Scholar 

  • Truesdell, C.A., ed. 1965b. Continuum Mechanics: Problems of Non-linear Elasticity, International Science Review Series, vol. VIII. New York: Gordon and Breach.

    Google Scholar 

  • Truesdell, C.A. 2012. Essays in the History of Mechanics. Berlin: Springer.

    Google Scholar 

  • Truesdell, C.A., and W. Noll. 1965. The Non-Linear Field Theories of Mechanics/Die Nicht-Linearen Feldtheorien der Mechanik in Handbook of Physics, vol. 3. Berlin: Springer.

    Book  Google Scholar 

  • Tucker, R. 2015. Rubber. In The Cambridge World History, ed. J. McNeill and K. Pomeranz, 423–443. Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Villaggio, P. 2011. Sixty years of solid mechanics. Meccanica 46: 1171–1189.

    Article  MathSciNet  Google Scholar 

  • Von Karman, T. 1940. The engineer grapples with nonlinear problems. Bulletin of the American Mathematical Society 46: 615–683.

    Article  MathSciNet  Google Scholar 

  • Wang, C.C., and C.A. Truesdell. 1973. Introduction to Rational Elasticity. Leyden: Noordhoff Int. Publishing.

    Google Scholar 

  • Zvolinskii, N.V., and P.M. Riz. 1939. On certain problems of the nonlinear theory of elasticity. Prikladnaya Matematika i Mekhanika 2: 4.

    Google Scholar 

Download references

Acknowledgements

Partially supported by “Gruppo Nazionale per la Fisica Matematica” (GNFM) of “Istituto Nazionale di Alta Matematica” (INDAM), Italy.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy

    Giuseppe Saccomandi

  2. School of Mathematics, Statistics and Applied Mathematics, NUI Galway, Galway, Ireland

    Giuseppe Saccomandi

  3. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Maurizio Stefano Vianello

Authors
  1. Giuseppe Saccomandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maurizio Stefano Vianello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurizio Stefano Vianello.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Communicated by Jesper Lützen.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saccomandi, G., Vianello, M.S. Antonio Signorini and the proto-history of the non-linear theory of elasticity. Arch. Hist. Exact Sci. 78, 375–400 (2024). https://doi.org/10.1007/s00407-024-00328-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00407-024-00328-2

Search

Navigation