Vibrations of elastic systems with a large number of tiny heavy inclusions

Article type: Research Article

Authors: Rybalko, Volodymyr

Affiliations: Laboratoire de Physique Mathématique et Géométrie, case 7012, Université Paris 7–Denis‐Diderot, 2 place Jussieu, 75251 Paris cedex 05, France E‐mail: vrybalko@ilt.kharkov.ua

Note: [] On leave from Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin ave., 61164, Kharkov, Ukraine.

Abstract: We consider a spectral problem modeling natural vibrations of a complex medium consisting of an elastic medium and tiny rigid inclusions. We study the asymptotic behaviour of the eigenvalues and eigenvectors of this problem when the total number of inclusions and their density tend to infinity. We obtain a limiting problem, that is a spectral problem for a linear fractional operator pencil, describing the macroscopic behaviour of the system (global vibrations). We also show that there exist vibrations of the medium localized in small vicinities of the inclusions (local vibrations), which correspond to eigenvalues accumulating at the poles of the operator pencil.

Keywords: homogenization, spectral analysis, concentrated masses

Journal: Asymptotic Analysis, vol. 32, no. 1, pp. 27-62, 2002