Geodesic
A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 152 (2007) no. 1, pp. 177-190 Cet article a éte moissonné depuis la source Math-Net.Ru

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Using a solitonic connection, we show that the class of infinitesimal Bäcklund transformations originally introduced by Loewner in 1952 in a gasodynamic context results in physically interesting nonlinear model constitutive laws. We obtain laws previously used to model a variety of hard and soft nonlinear elastic responses. A natural extension of the latter leads to a novel class of model constitutive laws where the stress and strain are given parametrically in terms of elliptic functions. Such models allow a change in the concavity of the stress–strain law. Such behavior can be observed in the compression of polycrystalline materials or in the unloading regimes of superelastic nickel–titanium.
Keywords: nonlinearity, elasticity, Loewner theory. 
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C. Rogers; W. K. Schief; K. W. Chow. A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 152 (2007) no. 1, pp. 177-190. http://geodesic.mathdoc.fr/item/TMF_2007_152_1_a13/

[1] C. Loewner, J. Anal. Math., 2 (1953), 219–242 | DOI | MR | Zbl

[2] H. M. Cekirge, E. Varley, Philos. Trans. R. Soc. Lond. Ser. A, 273 (1973), 261–313 | DOI | Zbl

[3] J. Y. Kazakia, E. Varley, Philos. Trans. R. Soc. Lond. Ser. A, 277 (1974), 191–237 | DOI | Zbl

[4] J. Y. Kazakia, E. Varley, Philos. Trans. R. Soc. Lond. Ser. A, 277 (1974), 239–250 | DOI | Zbl

[5] T. W. Duerig, A. R. Pelton, D. Stöckel, “The use of superelasticity in medicine”, Fachzeitschrift für Handel Wirtschaft, Technik und Wissenschaft, Sonderdruck aus Heft, 9/96, Metall Verlag-Hüthig GmbH, 1996, 569–574

[6] T. W. Duerig, “Present and future applications of shape memory and superelastic materials”, Mater. Res. Soc. Symp. Proc., 360, Materials Research Society, Pittsburgh, PA, 1995, 497–506 | DOI

[7] C. Loewner, Tech. Notes Nat. Adv. Comm. Aeronaut., 1950, no. 2065, 1–56 | MR

[8] W. K. Schief, C. Rogers, Stud. Appl. Math., 100 (1998), 391–422 | DOI | MR | Zbl

[9] C. Rogers, W. K. Schief, “Infinitesimal Bäcklund transformations of $K$-nets. The $2+1$-dimensional sinh-Gordon system”, Bäcklund and Darboux Transformations. The Geometry of Solitons (Halifax, Canada, 1999), CRM Proc. Lecture Notes, 29, Amer. Math. Soc., Providence, RI, 2001, 385–392 | DOI | MR | Zbl

[10] L. P. Eisenhart, Transformations of Surfaces, Chelsea, New York, 1962 | MR | Zbl

[11] B. G. Konopelchenko, C. Rogers, Phys. Lett. A, 158 (1991), 391–397 | DOI | MR

[12] B. G. Konopelchenko, C. Rogers, J. Math. Phys., 34 (1993), 214–242 | DOI | MR | Zbl

[13] O. Babelon, D. Bernard, Int. J. Mod. Phys. A, 8 (1993), 507–543 | DOI | MR | Zbl

[14] J. Dorfmeister, J. Inoguchi, M. Toda, “Weierstrass type representation of time like surfaces with constant mean curvature”, Differential Geometry and Integrable Systems (Tokyo, Japan, 2000), Contemp. Math., 308, Amer. Math. Soc., Providence, RI, 2002, 77–99 | DOI | MR | Zbl

[15] G. A. Dombrovskii, Metod approksimatsii adiabaty v teorii ploskikh techenii gaza, Nauka, M., 1964 | MR | Zbl

[16] G. Power, C. Rogers, R. A. Osborn, Z. Angew. Math. Mech., 49 (1969), 333–340 | DOI | Zbl

[17] C. Rogers, W. F. Shadwick, Bäcklund Transformations and Their Applications, Math. Sci. Engrg., 161, Academic Press, New York, 1982 | MR | Zbl

[18] J. F. Bell, The Physics of Large Deformations of Crystalline Solids, Springer, Berlin–New York–Heidelberg, 1968

[19] H. M. Cekirge, C. Rogers, Arch. Mech., 29 (1977), 125–141 | MR | Zbl