Open AccessWave dispersion curves in discrete lattices derived through asymptotic multi-scale method
Author(s) -
Antoine Rallu,
Claude Boutin,
Stéphane Hans
Publication year - 2019
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1264/1/012012
Subject(s) - homogenization (climate) , dispersion (optics) , computation , amplitude , mathematical analysis , asymptotic homogenization , lattice (music) , diffraction , scale (ratio) , wave propagation , dispersion relation , mathematics , vibration , physics , algorithm , optics , acoustics , biodiversity , ecology , quantum mechanics , composite number , biology
This paper falls within the study of dispersion feature of elastic periodic media. In most cases, no analytic description is reachable and the problem is solved via numerical computations of the dispersion curves.We propose in this paper an analytic method dedicated to lattice systems that enables to reconstruct part by part the dispersion curves via an asymptotic multi-scale method. This method is illustrated on periodic reticulated beams. At low frequency, when there is a large scale separation between the length of the cell and the characteristic size of the vibrations, the classical homogenization method allows efficiently to establish the continuous equivalent dynamic description and the associated wave propagation properties. This scale separation is lost for frequencies of the order or higher than the diffraction frequency. However, instead of considering the amplitude of the mean displacement in a unit cell, the concept of scale separation can still be used by considering the amplitude of periodic eigenmodes defined on (multi-)cells. Thus, similar principles of asymptotic multi-scale method enables to describe the large scale modulations around the eigenfrequencies of the mono-and/or multi-cells period. Finally, the properties of the modulation are straightforwardly related to the dispersion curves at the considered frequencies.