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

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.