Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials

Abstract

The work is devoted to the analysis of the spectral properties of a boundary value problem describing one-dimensional vibrations along the axis O_x1 of periodically alternating M elastic and M viscoelastic layers parallel to the plane O_x2x3. It is shown that the spectrum of the boundary value problem is the union of roots of M equations. The asymptotic behavior of the spectrum of the problem as M → ∞ is analyzed; in particular, it is proved that not all sequences of eigenvalues of the original (prelimit) problem converge to eigenvalues of the corresponding homogenized (limit) problem.