Trudy Matematicheskogo Instituta imeni V.A. Steklova
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy Mat. Inst. Steklova: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (324 kB)
References: PDF file   HTML file

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

Bibliographic databases:
UDC: 539.3+534.1

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 

• http://mi.mathnet.ru/eng/tm49
• http://mi.mathnet.ru/eng/tm/v251/p173

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. S. K. Godunov, I. M. Peshkov, “Symmetric hyperbolic equations in the nonlinear elasticity theory”, Comput. Math. Math. Phys., 48:6 (2008), 975–995
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
•  Number of views: This page: 452 Full text: 139 References: 65