Vol. 4, No. 5, 2009

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ISSN: 1559-3959
A dispersive nonlocal model for wave propagation in periodic composites

Juan Miguel Vivar-Pérez, Ulrich Gabbert, Harald Berger, Reinaldo Rodríguez-Ramos, Julián Bravo-Castillero, Raul Guinovart-Díaz and Federico J. Sabina

Vol. 4 (2009), No. 5, 951–976
Abstract

In this paper, the problem of wave propagation in periodic structured composites is studied, and a dispersive asymptotic method for the description of these dynamic processes is proposed. Assuming a single-frequency dependence of the solution for the one dimensional wave equation in a periodic composite material, higher-order terms in the asymptotic expansion for the displacement functions are studied. Nonuniformity is eliminated by finding a suitable regular asymptotic expansion for the perturbation frequency. Only two spatial scales are considered, and the equivalence of this method and the introduction of multiple slow temporal scales is shown, in good agreement with previous approaches. For a selection of boundary problems, analytic solutions are given and graphically illustrated. The problem of failures is also discussed, and some illustrative calculations are presented.

Keywords
composite materials, wave dispersion, asymptotics, homogenization, wave propagation, bilaminated composite, dynamic asymptotic homogenization
Milestones
Received: 8 March 2009
Accepted: 17 May 2009
Published: 5 September 2009
Authors
Juan Miguel Vivar-Pérez
Facultad de Matemática y Computación
Universidad de La Habana
San Lázaro esq. L, Vedado
Habana 4
CP 10400
Cuba
Ulrich Gabbert
Facultät für Maschinembau
Otto-von-Guericke Universität
Universitätsplatz 2
39106 Magdeburg
Germany
Harald Berger
Facultät für Maschinembau
Otto-von-Guericke Universität
Universitätsplatz 2
39106 Magdeburg
Germany
Reinaldo Rodríguez-Ramos
Facultad de Matemática y Computación
Universidad de La Habana
San Lázaro esq. L, Vedado
Habana 4
CP 10400
Cuba
Julián Bravo-Castillero
Facultad de Matemática y Computación
Universidad de La Habana
San Lázaro esq. L, Vedado
Habana 4
CP 10400
Cuba
Raul Guinovart-Díaz
Facultad de Matemática y Computación
Universidad de La Habana
San Lázaro esq. L, Vedado
Habana 4
CP 10400
Cuba
Federico J. Sabina
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas
Universidad Nacional Autónoma de México
Apartado Postal 20-726
Delegación de Álvaro Obregón
01000 México, DF
Mexico