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Research Papers

Wave Propagation in Periodic Composites: Higher-Order Asymptotic Analysis Versus Plane-Wave Expansions Method

[+] Author and Article Information
I. V. Andrianov

Institut für Allgemeine Mechanik, RWTH Aachen University, Templergraben 64, D-52056 Aachen, Germanyigor_andrianov@inbox.ru

J. Awrejcewicz

Department of Automatics and Biomechanics, Technical University of Łódź, 1/15 Stefanowski Street, 90-924 Łódź, Polandawrejcew@p.lodz.pl

V. V. Danishevs’kyy

 Prydniprovs’ka State Academy of Civil Engineering and Architecture, 24a Chernishevskogo Street, 49600 Dnipropetrovs’k, Ukrainevdanish@ukr.net

D. Weichert

Institut für Allgemeine Mechanik, RWTH Aachen University, Templergraben 64, D-52056 Aachen, Germanyweichert@iam.rwth-aachen.de

J. Comput. Nonlinear Dynam 6(1), 011015 (Oct 07, 2010) (8 pages) doi:10.1115/1.4002389 History: Received May 25, 2009; Revised July 30, 2010; Published October 07, 2010; Online October 07, 2010

This work is devoted to a comparison of different methods determining stop-bands in 1D and 2D periodic heterogeneous media. For a 1D case, the well-known dispersion equation is studied via asymptotic approach. In particular, we show how homogenized solutions can be obtained by elementary series used up to any higher-order. We illustrate and discuss a possible application of asymptotic series regarding parameters other than wavelength and frequency. In addition, we study antiplane elastic shear waves propagating in the plane through a spatially infinite periodic composite material consisting of an infinite matrix and a square lattice of circular inclusions. In order to solve the problem, a homogenization method matched with asymptotic solution on the cell with inclusion of the large volume fracture is proposed and successfully applied. First and second approximation terms of the averaging method provide the estimation of the first stop-band. For validity and comparison with other approaches, we have also applied the Fourier method. The Fourier method is shown to work well for relatively small inclusions, i.e., when the inclusion-associated parameters and matrices slightly differ from each other. However, for evidently contrasting structures and for large inclusions, a higher-order homogenization method is advantageous. Therefore, a higher-order homogenization method and the Fourier analysis can be treated as mutually complementary.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

1D composite material (L1+L2=d)

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Figure 2

Dispersion curves in the first pass band for the steel-aluminum 1D composite

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Figure 7

Dispersion curves in the first pass band for the carbon-epoxy composite

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Figure 6

Dispersion curves in the first pass band for the nickel-aluminum 2D composite

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Figure 5

(a) Phononic bands for the nickel-aluminum composite (dashed line: jmax=1 and solid line: jmax=2) and (b) directions of the wave vector μ

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Figure 4

A periodically repeated unit cell

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Figure 3

Composite structure under consideration (l is the size of the unit cell and λ is the wavelength of the traveling signal)

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