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Saccomandi, Giuseppe:
Il Problema Centrale della Teoria dell'Elasticità Non-Lineare Secondo Signorini
Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana Serie 1 9 (2024), fasc. n.1, p. 5-27, (Italian)
pdf (1.29 MB), djvu (546 Kb). | Zbl 07958490

Sunto

Nel 1942 Antonio Signorini, parlando della teoria dell'elasticità non-lineare durante un convegno internazionale presso l'INDAM, dichiara che nell'ambito di questa teoria:
“... 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.”
Signorini pensava a calcoli numerici che potessero indicare la teoria dell'elasticità non-lineare come un modello matematico adeguato a descrivere il comportamento di alcuni materiali. In questa nota si illustra come il problema di Signorini dal 1942 a oggi abbia attratto l'attenzione non solo dei meccanici ma anche di molti matematici. Si illustrano anche da un punto di vista storico i principali risultati ottenuti in questi ottanta anni e si indicano i problemi fisico-matematici che sono ancora aperti..
Referenze Bibliografiche
[1] ALMANSI, E. (1911). Sulle deformazioni finite dei solidi elastici isotropi. Rendiconti della Reale Accademia dei Lincei: Classe di scienze fisiche, matematiche e naturali (Vol. 20, pp. 705-714). | Zbl 42.0865.04
[2] ARMANNI, G. (1915). Sulle deformazioni finite dei soldi elastici isotropi. Il Nuovo Cimento (1911-1923), 9(1), 427-447. | Zbl 45.1037.03
[3] ANSSARI-BENAM, A., BUCCHI, A., DESTRADE, M., & SACCOMANDI, G. (2022). The Generalised Mooney Space for Modelling the Response of Rubber-Like Materials. Journal of Elasticity, 1-15. | fulltext (doi) | MR 4496447 | Zbl 1511.74002
[4] ANTMAN, S. S. (1995) Nonlinear Problems of Elasticity. Springer, New York, NY. | fulltext (doi) | MR 1323857 | Zbl 0820.73002
[5] ARRUDA, E. M., & BOYCE, M. C. (1993). A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41(2), 389-412. | Zbl 1355.74020
[6] BALL, J. M. (1976). Convexity conditions and existence theorems in nonlinear elasticity. Archive for rational mechanics and Analysis, 63(4), 337-403. | fulltext (doi) | MR 475169 | Zbl 0368.73040
[7] BALL, J. M., & JAMES, R. D. (2002). The Scientific Life and Influence of Clifford Ambrose Truesdell III. Archive for Rational Mechanics and Analysis, 161(1), 1-26. | fulltext (doi) | MR 1883756 | Zbl 0988.01007
[8] BAKER, G.A. & GRAVES-MORRIS P. (1996): Padé Approximants Encyclopedia of Mathematics and its Applications CUP. | fulltext (doi) | MR 1383091 | Zbl 0923.41001
[9] BELLINO, L., FLORIO, G., GORIELY, A. & PUGLISI, G. (2023) Cooperative melting in double-stranded peptide chains through local mechanical interactions. J. R. Soc. Interface, 20:2023013020230130. | Zbl 0871.90018
[10] BLAYSAT, B., BALANDRAUD, X., GRÉDIAC, M., VIVES, E., BARRERA, N., & ZANZOTTO, G. (2020). Concurrent tracking of strain and noise bursts at ferroelastic phase fronts. Communications Materials, 1(1), 3. | Zbl 07867736
[11] BRILLOUIN, L. (1925) Sur les tensions de radiation. Ann.Phys., 4:528-86. | Zbl 51.0648.05
[12] DESTRADE, M., & SACCOMANDI, G. (2010). On the rectilinear shear of compressible and incompressible elastic slabs. International journal of engineering science,48(11), 1202-1211. | fulltext (doi) | MR 2760979 | Zbl 1231.74041
[13] DESTRADE, M., GILCHRIST, M. D., MURPHY, J. G., RASHID, B., & SACCOMANDI, G. (2015). Extreme softness of brain matter in simple shear. International Journal of NonLinear Mechanics, 75, 54-58. | Zbl 07280065
[14] DESTRADE, M., SACCOMANDI, G., & SGURA, I. (2017). Methodical fitting for mathematical models of rubber-like materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2198), 20160811. | Zbl 1404.74016
[15] DESTRADE, M., MURPHY, J., & SACCOMANDI, G. (2019). Rivlin's legacy in continuum mechanics and applied mathematics. Philosophical Transactions of the Royal Society A, 377(2144), 20190090. | fulltext (doi) | MR 3947229 | Zbl 1425.74007
[16] DESTRADE, M., DORFMANN, L., & SACCOMANDI, G. (2022).The Ogden model of rubber mechanics: 50 years of impact on nonlinear elasticity. Philosophical Transactions of the Royal Society A, 380(2234), 20210332. | MR 4498076 | Zbl 1326.01052
[17] DE TOMMASI, D., PUGLISI, G., & SACCOMANDI, G. (2015). Multiscale mechanics of macromolecular materials with unfolding domains. Journal of the Mechanics and Physicsof Solids, 78, 154-172. | fulltext (doi) | MR 3349456 | Zbl 1349.74285
[18] ERICKSEN, J. L. (1954). Deformations possible in every isotropic, incompressible, perfectly elastic body. Zeitschrift für angewandte Mathematik und Physik ZAMP, 5(6),466-489. | fulltext (doi) | MR 66873 | Zbl 0059.17509
[19] ERICKSEN, J. L. (1955). Deformations possible in every compressible, isotropic, perfectly elastic material. Journal of Mathematics and Physics, 34(1-4), 126-128. | fulltext (doi) | MR 70397 | Zbl 0064.42105
[20] FICHERA, G. (1979). Il contributo italiano alla teoria matematica dell'elasticità. Rendiconti del Circolo Matematico di Palermo, 28(1), 5-26. | Zbl 0433.73002
[21] GENT, A.N. (1996) A New Constitutive Relation for Rubber. Rubber Chemistry and Technology: March 1996, Vol. 69, No. 1, pp. 59-61. | Zbl 1349.74053
[22] GRIOLI, G. (1962). Mathematical theory of elastic equilibrium: recent results (ERG ANGEW MATHE, volume 7). Springer Science & Business Media. Chicago. | MR 154463 | Zbl 0102.17004
[23] HASTINGS, S. P., & MCLEOD, J. B. (2011). Classical methods in ordinary differential equations: with applications to boundary value problems (Vol. 129). American Mathematical Soc. | fulltext (doi) | MR 2865597 | Zbl 1239.34001
[24] HORGAN, C. O. (2015). The remarkable Gent constitutive model for hyperelastic materials. International Journal of Non-Linear Mechanics, 68, 9-16. | Zbl 1384.03038
[25] HORGAN, C. O., & SACCOMANDI, G. (1999). Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility. Journal of Elasticity, 56(2), 159-170. | fulltext (doi) | MR 1799932 | Zbl 0982.74011
[26] HORGAN, C. O., & SACCOMANDI, G. (2002). A molecular statistical basis for the Gent constitutive model of rubber elasticity. Journal of Elasticity, 68(1), 167-176. | fulltext (doi) | MR 2024310 | Zbl 1073.74007
[27] HUMPHREY, J. D. (1995). Mechanics of the arterial wall: review and directions. Critical Reviews in Biomedical Engineering, 23(1-2). | Zbl 0833.01029
[28] KEARSLEY, E. A. (1989). Note: Strain invariants expressed as average stretches. Journal of Rheology, 33(5), 757-760. | Zbl 1253.35107
[29] KLEIMAN, S. L. (1997). Bertini and his two fundamental theorems. ArXiv preprint alg-geom/9704018. | MR 1661859 | Zbl 0926.14001
[30] MANGAN, R., DESTRADE, M., & SACCOMANDI, G. (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. | Zbl 1231.74041
[31] MARCOLONGO, R. (1907). Progressi e sviluppo della teoria matematica della elasticità in Italia (1870-1907). Il Nuovo Cimento (1901-1910), 14(1), 371-393. | Zbl 38.0070.02
[32] MOONEY, M. (1940). A theory of large elastic deformation. Journal of applied physics, 11(9), 582-592. | Zbl 66.1021.04
[33] OGDEN, R. W. (1972). Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubber like solids. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 326(1567), 565-584. | Zbl 0257.73034
[34] OGDEN, R. W., SACCOMANDI, G., & SGURA, I. (2004). Fitting hyperelastic models to experimental data. Computational Mechanics, 34, 484-502. | Zbl 1109.74320
[35] PODIO-GUIDUGLI, P., & CAFFARELLI, G. V. (1991). Extreme elastic deformations. Archive for rational mechanics andanalysis, 115, 311-328. | fulltext (doi) | MR 1120851 | Zbl 0784.73007
[36] PUCCI, E., & SACCOMANDI, G. (1997). On universal relations in continuum mechanics. Continuum Mechanics and Thermodynamics, 9(2), 61-72. | fulltext (doi) | MR 1465825 | Zbl 0873.73012
[37] PUCCI, E., & SACCOMANDI, G. (2002). A note on the Gent model for rubber-like materials. Rubber chemistry and technology, 75(5), 839-852. | Zbl 1045.34014
[38] PUCCI, E., & SACCOMANDI, G. (2013). The anti-plane shear problem in nonlinear elasticity revisited. Journal of Elasticity, 113(2), 167-177. | fulltext (doi) | MR 3102593 | Zbl 1345.74017
[39] PUCCI, E., RAJAGOPAL, K. R., & SACCOMANDI, G. (2015). On the determination of semi-inverse solutions of nonlinear Cauchy elasticity: The not so simple case of anti-plane shear. International Journal of Engineering Science, 88, 3-14. | fulltext (doi) | MR 3306530 | Zbl 1423.74136
[40] PUCCI, E., SACCOMANDI, G., & VITOLO, R. (2016). Bogus transformations in mechanics of continua. International Journal of Engineering Science, 99, 13-21. | fulltext (doi) | MR 3434387 | Zbl 1423.74137
[41] PUGLISI, G., & SACCOMANDI, G. (2015). The Gent model for rubber-like materials: an appraisal for an ingenious and simple idea. International Journal of Non-Linear Mechanics, 68, 17-24. | Zbl 1349.74285
[42] PUGLISI, G., & SACCOMANDI, G. (2016). Multi-scale modelling of rubber-like materials and soft tissues: an appraisal. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2187), 20160060. | Zbl 1349.74285
[43] QUINTANILLA, R., & SACCOMANDI, G. (2007). The importance of the compatibility of nonlinear constitutive theories with their linear counterparts. J. Appl. Mech., 74(3):455-460 | fulltext (doi) | MR 2308820 | Zbl 1111.74605
[44] RIVLIN, R. S. (1948). Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philosophical transactions of the royal society of London. Series A, Mathematical and physical sciences, 241(835), 379-397. | fulltext (doi) | MR 27674 | Zbl 0031.42602
[45] SACCOMANDI, G. (2001). Universal results in finite elasticity, Nonlinear elasticity: theory and applications, volume 283 of London Math. Soc. Lecture Note Ser., 97-134. | fulltext (doi) | MR 1835110 | Zbl 0993.74009
[46] SACCOMANDI, G., & VERGORI, L. (2021). Some Remarks on the Weakly Nonlinear Theory of Isotropic Elasticity. Journal of Elasticity, 1-26. | fulltext (doi) | MR 4368610 | Zbl 1481.74076
[47] SACCOMANDI, G., VERGORI, L., & ZANETTI, E. M. (2022). Linear, weakly nonlinear and fully nonlinear models for soft tissues: which ones provide the most reliable estimations of the stiffness? Philosophical Transactions of the Royal Society A, 380(2234), 20210321. | MR 4498090 | Zbl 1481.74076
[48] SACCOMANDI, G., & VIANELLO, M. S. (2024) Antonio Signorini and the proto-history of the non-linear theory of elasticity. Arch. Hist. Exact Sci. (2024). https://doi.org/10.1007/s00407-024-00328-2. | fulltext (doi) | MR 4756156 | Zbl 1545.01030
[49] SIGNORINI, A. (1943). Trasformazioni termoelastiche finite. Annali di Matematica pura ed applicata, 22, 33-143. | fulltext (doi) | MR 18090 | Zbl 0102.17607
[50] SIGNORINI, A. (1945) Recenti progressi della teoria delle trasformazioni termoelastiche finite in Atti del Convegno matematico del 1942 Roma, Bardi, 153-168. 31, 35-53. | MR 21838 | Zbl 0061.42801
[51] SIGNORINI, A. (1949). Trasformazioni termoleastiche finite. Annali di Matematica Pura ed Applicata, 30, 1-72. | fulltext (doi) | MR 35616 | Zbl 0041.56101
[52] SIGNORINI, A. (1961) Risultati semplici della teoria non linearizzata dell'Elasticita. Rendiconti del Seminario Matematico e Fisico di Milano 31, 35-53. | MR 127586 | Zbl 0103.17503
[53] TAYLOR, G. I. (1950) Seventh International Congress for Applied Mechanics, 1948. (1950) Nature. | Zbl 1001.68750
[54] TRELOAR, L. R. G. (1949) Physics of Rubber Elasticity. OUP. | Zbl 0347.73042
[55] TRUESDELL, C. (1956). Das ungelöste Hauptproblem der endlichen Elastizitätstheorie. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 36(3-4), 97-103. | fulltext (doi) | MR 79892 | Zbl 0070.41703
[56] WANG, C. C., & TRUESDELL, C. (1973). Introduction to rational elasticity (Vol. 1). Springer Science & Business Media. | MR 468442 | Zbl 0308.73001
[57] YAVARI, A., & GORIELY, A. (2023). The universal program of nonlinear hyperelasticity. Journal of Elasticity, 154(1), 91-146. | fulltext (doi) | MR 4661777 | Zbl 1528.74015

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