2010/12/26 by A. L. Shuvalov, A. A. Kutsenko, Anton A. Kutsenko +4
Engineering · Mathematics · Physics and Astronomy · #Acoustic Wave Phenomena Research #Railway Engineering and Dynamics #Vibration and Dynamic Analysis #cond-mat.mtrl-sci #math-ph #math.MP
paper · pdf · doi:10.1098/rspa.2010.0389
published as Proc. R. Soc. A, 467, 1749-1769, 2011 · 20 pages, 3 figures
arxiv created 2010/12/26 · openalex publication_date 2011/01/13 · arxiv updated 2011/04/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is exemplified for scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series, and explicit estimates are derived for the rate of convergence.