[Vibrations de systèmes élastiques avec un grand nombre de petites inclusions à forte densité]
On considère un problème spectral qui modélise les vibrations propres d'un milieu complexe constitué d'un milieu élastique et d'un grand nombre de petites inclusions rigides à forte densité. On étudie le comportement asymptotique de ce problème lorsque le nombre d'inclusions et leur densité tendent vers l'infini. On obtient un problème spectral limite pour une famille rationnelle fractionnaire d'opérateurs qui décrit le comportement macroscopique du système (vibrations globales).
We consider a spectral problem modeling natural vibrations of a complex medium that consists of an elastic medium and tiny rigid inclusions. We study the asymptotic behaviour of solutions of this problem when the total number of inclusions and their density tend to infinity. We obtain a limit problem being a spectral problem for a linear fractional operator pencil that describes the macroscopic behaviour of the system (global vibrations).
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@article{CRMATH_2002__334_3_245_0,
author = {Rybalko, Volodymyr},
title = {Vibrations of elastic systems with a large number of tiny heavy inclusions},
journal = {Comptes Rendus. Math\'ematique},
pages = {245--250},
publisher = {Elsevier},
volume = {334},
number = {3},
year = {2002},
doi = {10.1016/S1631-073X(02)02233-1},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02233-1/}
}
TY - JOUR AU - Rybalko, Volodymyr TI - Vibrations of elastic systems with a large number of tiny heavy inclusions JO - Comptes Rendus. Mathématique PY - 2002 SP - 245 EP - 250 VL - 334 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02233-1/ DO - 10.1016/S1631-073X(02)02233-1 LA - en ID - CRMATH_2002__334_3_245_0 ER -
%0 Journal Article %A Rybalko, Volodymyr %T Vibrations of elastic systems with a large number of tiny heavy inclusions %J Comptes Rendus. Mathématique %D 2002 %P 245-250 %V 334 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02233-1/ %R 10.1016/S1631-073X(02)02233-1 %G en %F CRMATH_2002__334_3_245_0
Rybalko, Volodymyr. Vibrations of elastic systems with a large number of tiny heavy inclusions. Comptes Rendus. Mathématique, Tome 334 (2002) no. 3, pp. 245-250. doi : 10.1016/S1631-073X(02)02233-1. http://www.numdam.org/articles/10.1016/S1631-073X(02)02233-1/
[1] Vibrations of a body with many concentrated masses near the boundary: high frequency vibrations, Spectral Analysis of Complex Structures (Paris, 1993), Hermann, Paris, 1995, pp. 85-101
[2] On vibrations of a body with many concentrated masses near the boundary, Math. Models Methods Appl. Sci., Volume 3 (1993) no. 2, pp. 249-273
[3] Vibrations of a membrane with many concentrated masses near the boundary, Math. Models Methods Appl. Sci., Volume 5 (1995) no. 5, pp. 565-585
[4] The Integral Equations of the Theory of Elasticity, Teubner-Texte zur Mathematik, Teubner, Stuttgart, 1995
[5] The Neumann sieve, Nonlinear Variational Problems (Isola d'Elba, 1983), Res. Notes Math., 127, Pitman, Boston, 1985, pp. 24-32
[6] Frequencies of natural oscillations of bodies with concentrated masses, Functional and Numerical Methods in Mathematical Physics, 271, Naukova Dumka, Kiev, 1988, pp. 165-171 (in Russian)
[7] Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), Lecture Notes in Phys., 195, Springer, Berlin–New York, 1984, pp. 346-368
[8] Vibration de systèmes élastiques avec des masses concentrées, Rend. Sem. Mat. Univ. Politec. Torino, Volume 42 (1984) no. 3, pp. 43-63
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