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Trudy Mat. Inst. Steklova, 2005, Volume 251, Pages 173–199 (Mi tm49)  

This article is cited in 3 scientific papers (total in 3 papers)

Some Problems in Nonlinear Dynamic Elasticity

A. G. Kulikovskiia, E. I. Sveshnikovab, A. P. Chugainovaa

a Steklov Mathematical Institute, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper contains a survey of recent studies of small-amplitude quasi-transverse one-dimensional waves in elastic and viscoelastic media. The following issues are addressed: We study small-amplitude nonlinear waves in elastic media under a more accurate consideration of the internal energy compared with the earlier works. We describe new properties of shock waves and Riemann waves of small amplitude in an anisotropic medium whose properties are invariant under the rotation through $120^\circ $ about the wave normal. We formulate similarity conditions for one-dimensional problems of nonlinear elasticity. We discuss reasons for the earlier discovered nonuniqueness of solutions to self-similar problems for waves in elastic media, and formulate a criterion that allows one to predict, based solely on the properties of the shock adiabat, the nonuniqueness or the nonexistence of self-similar solutions to systems of hyperbolic equations that express conservation laws. We consider the structure of shock waves in elastic media in the framework of the Kelvin–Voigt model of a viscous medium. The results of the numerical analysis of the nonlinear stability of the structure of metastable shock waves are also presented.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 251, 165–191

Bibliographic databases:
UDC: 539.3+534.1
Received in September 2004

Citation: A. G. Kulikovskii, E. I. Sveshnikova, A. P. Chugainova, “Some Problems in Nonlinear Dynamic Elasticity”, Nonlinear dynamics, Collected papers, Trudy Mat. Inst. Steklova, 251, Nauka, MAIK Nauka/Inteperiodika, M., 2005, 173–199; Proc. Steklov Inst. Math., 251 (2005), 165–191

Citation in format AMSBIB
\Bibitem{KulSveChu05}
\by A.~G.~Kulikovskii, E.~I.~Sveshnikova, A.~P.~Chugainova
\paper Some Problems in Nonlinear Dynamic Elasticity
\inbook Nonlinear dynamics
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2005
\vol 251
\pages 173--199
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm49}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2234381}
\zmath{https://zbmath.org/?q=an:1138.74349}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2005
\vol 251
\pages 165--191


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. G. Kulikovskii, A. P. Chugainova, “Classical and Nonclassical Discontinuities and Their Structure in Nonlinear Elastic Media with Dispersion and Dissipation”, Proc. Steklov Inst. Math., 276, suppl. 2 (2012), S1–S68  mathnet  crossref  crossref  zmath  isi
    2. S. K. Godunov, I. M. Peshkov, “Symmetric hyperbolic equations in the nonlinear elasticity theory”, Comput. Math. Math. Phys., 48:6 (2008), 975–995  mathnet  crossref  zmath  isi  elib  elib
    3. Kulikovskii A.G., Sveshnikova Y.I., “A model for describing near–resonance oscillations in an elastic layer”, Pmm Journal of Applied Mathematics and Mechanics, 72:6 (2008), 715–723  crossref  mathscinet  adsnasa  isi  scopus
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