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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awrejcew@p.lodz.pl
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January 2011
Research Papers
Wave Propagation in Periodic Composites: Higher-Order Asymptotic Analysis Versus Plane-Wave Expansions Method
I. V. Andrianov,
I. V. Andrianov
Institut für Allgemeine Mechanik,
igor_andrianov@inbox.ru
RWTH Aachen University
, Templergraben 64, D-52056 Aachen, Germany
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J. Awrejcewicz,
J. Awrejcewicz
Department of Automatics and Biomechanics,
awrejcew@p.lodz.pl
Technical University of Łódź
, 1/15 Stefanowski Street, 90-924 Łódź, Poland
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V. V. Danishevs’kyy,
vdanish@ukr.net
V. V. Danishevs’kyy
Prydniprovs’ka State Academy of Civil Engineering and Architecture
, 24a Chernishevskogo Street, 49600 Dnipropetrovs’k, Ukraine
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D. Weichert
D. Weichert
Institut für Allgemeine Mechanik,
weichert@iam.rwth-aachen.de
RWTH Aachen University
, Templergraben 64, D-52056 Aachen, Germany
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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. Jan 2011, 6(1): 011015 (8 pages)
Published Online: October 7, 2010
Article history
Received:
May 25, 2009
Revised:
July 30, 2010
Online:
October 7, 2010
Published:
October 7, 2010
Citation
Andrianov, I. V., Awrejcewicz, J., Danishevs’kyy, V. V., and Weichert, D. (October 7, 2010). "Wave Propagation in Periodic Composites: Higher-Order Asymptotic Analysis Versus Plane-Wave Expansions Method." ASME. J. Comput. Nonlinear Dynam. January 2011; 6(1): 011015. https://doi.org/10.1115/1.4002389
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