Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

On ill-posedness of free-boundary problems for highly compressible two-dimensional elastic bodies


Authors: Yu. V. Egorov and E. Sanchez-Palencia
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 913-926
MSC (2010): Primary 35R25; Secondary 74B99
Published electronically: August 18, 2011
MathSciNet review: 2760087
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Some problems of elasticity theory related to highly compressible two-dimensional elastic bodies are considered. Such problems arise in real elasticity and pertain to some materials having negative Poisson ratio. The common feature of such problems is the presence of a small parameter $ \varepsilon$. If $ \varepsilon>0$, the corresponding equations are elliptic and the boundary data obey the Shapiro-Lopatinsky condition. If $ \varepsilon=0$, this condition is violated and the problem may fail to be solvable in distribution spaces. The rather difficult passing to the limit is studied.


References [Enhancements On Off] (What's this?)

  • 1. Yu. V. Egorov and M. A. Shubin, Linear partial differential equations. Foundations of the classical theory, Partial differential equations, 1 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 5–265 (Russian). MR 1141629
    Yu. V. Egorov and M. A. Shubin, Foundations of the classical theory of partial differential equations, Springer-Verlag, Berlin, 1998. Translated from the 1988 Russian original by R. Cooke; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Partial differential equations. I, Encyclopaedia Math. Sci., 30, Springer, Berlin, 1992; MR1141630 (93a:35004b)]. MR 1657445
  • 2. L. D. Landau and E. M. Lifshits, Teoreticheskaya fizika. Tom VII, 4th ed., “Nauka”, Moscow, 1987 (Russian). Teoriya uprugosti. [Theory of elasticity]; Edited by Lifshits, A. M. Kosevich and L. P. Pitaevskiĭ. MR 912888
    L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 7, 3rd ed., Pergamon Press, Oxford, 1986. Theory of elasticity; Translated from the Russian by J. B. Sykes and W. H. Reid. MR 884707
  • 3. F. Béchet, E. Sanchez-Palencia, and O. Millet, Singular perturbations generating complexification phenomena for elliptic shells, Comput. Mech. 43 (2008), no. 2, 207–221. MR 2453540, https://doi.org/10.1007/s00466-008-0297-8
  • 4. Yuri V. Egorov, Nicolas Meunier, and Evariste Sanchez-Palencia, Rigorous and heuristic treatment of certain sensitive singular perturbations, J. Math. Pures Appl. (9) 88 (2007), no. 2, 123–147 (English, with English and French summaries). MR 2348766, https://doi.org/10.1016/j.matpur.2007.04.010
  • 5. Yuri V. Egorov, Nicolas Meunier, and Evariste Sanchez-Palencia, Rigorous and heuristic treatment of sensitive singular perturbations arising in elliptic shells, Around the research of Vladimir Maz’ya. II, Int. Math. Ser. (N. Y.), vol. 12, Springer, New York, 2010, pp. 159–202. MR 2676173, https://doi.org/10.1007/978-1-4419-1343-2_7
  • 6. R. F. Almgren, An anisotropic $ 3$-dimensional structure with Poisson ration-1, J. Elasticity 15 (1985), 427-430.
  • 7. R. Lakes, Foam structures with negative Poisson ratio, Science AAAS 235 (1987), 1038-1040.
  • 8. V. M. Babich and V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln. Tom l, Izdat. “Nauka”, Moscow, 1972 (Russian). Metod etalonnykh zadach. [The method of canonical problems]; With the collaboration of M. M. Popov and I. A. Molotkov. MR 0426630
    V. M. Babič and V. S. Buldyrev, Short-wavelength diffraction theory, Springer Series on Wave Phenomena, vol. 4, Springer-Verlag, Berlin, 1991. Asymptotic methods; Translated from the 1972 Russian original by E. F. Kuester. MR 1245488

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35R25, 74B99

Retrieve articles in all journals with MSC (2010): 35R25, 74B99


Additional Information

Yu. V. Egorov
Affiliation: Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, Toulouse 31062, France
Email: egorov@cegetel.net

E. Sanchez-Palencia
Affiliation: Laboratoire de Modélisation en Méchanique 4, Université Pierre et Marie Curie, place Jussieu, case 162, Paris 75252, France
Email: sanchez@lmm.jussieu.fr

DOI: https://doi.org/10.1090/S1061-0022-2011-01176-8
Keywords: Two-dimensional elasticity, negative Poisson ratio, elliptic boundary value problems
Received by editor(s): June 29, 2010
Published electronically: August 18, 2011
Dedicated: To V.M.Babich on the occasion of his 80th birthday
Article copyright: © Copyright 2011 American Mathematical Society