Advances in Materials Science and Engineering

Volume 2016, Article ID 3017835, 9 pages

http://dx.doi.org/10.1155/2016/3017835

## Plane Wave-Perturbative Method for Evaluating the Effective Speed of Sound in 1D Phononic Crystals

^{1}Benemérita Universidad Autónoma de Puebla, Ciudad Universitaria, Blvd. Valsequillo y Esquina, Av. San Claudio s/n, Col. San Manuel, 72570 Puebla, PUE, Mexico^{2}Instituto Tecnológico de Puebla, División de Estudios de Posgrado e Investigación, Av. Tecnológico No. 420, Maravillas, 72220 Puebla, PUE, Mexico^{3}CONACYT, Dirección Adjunta al Desarrollo Científico, Dirección de Cátedras, Insurgentes Sur 1582, Crédito Constructor, Benito Juárez, 03940 Ciudad de México, Mexico

Received 6 July 2016; Revised 1 October 2016; Accepted 3 November 2016

Academic Editor: Tiejun Liu

Copyright © 2016 J. Flores Méndez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A method for calculating the effective sound velocities for a 1D phononic crystal is presented; it is valid when the lattice constant is much smaller than the acoustic wave length; therefore, the periodic medium could be regarded as a homogeneous one. The method is based on the expansion of the displacements field into plane waves, satisfying the Bloch theorem. The expansion allows us to obtain a wave equation for the amplitude of the macroscopic displacements field. From the form of this equation we identify the effective parameters, namely, the effective sound velocities for the transverse and longitudinal macroscopic displacements in the homogenized 1D phononic crystal. As a result, the explicit expressions for the effective sound velocities in terms of the parameters of isotropic inclusions in the unit cell are obtained: mass density and elastic moduli. These expressions are used for studying the dependence of the effective, transverse and longitudinal, sound velocities for a binary 1D phononic crystal upon the inclusion filling fraction. A particular case is presented for 1D phononic crystals composed of W-Al and Polyethylene-Si, extending for a case solid-fluid.

#### 1. Introduction

At present there is a great interest in fabricating artificial materials, having extraordinary properties, which considerably extend those of natural materials. Such a new class of materials is known as metamaterials. From the beginning, the photonic metamaterials, possessing negative index of refraction, have intensively been investigated. It was established that such an unusual optical property is found in periodic structures (photonic crystals) whose dielectric function is spatially modulated. A peculiarity of the photonic metamaterials is the high dielectric contrast between the components. The negative refraction has been observed in double negative metamaterials with simultaneously negative effective permittivity and permeability [1–6]. However, this phenomenon is also manifested in anisotropic materials, as simple as a one-dimensional (1D) photonic crystal or superlattice, having elements in the effective permittivity tensor of different sign [7–10]. Analogously, metamaterials with uncommon acoustic properties have also searched among phononic crystals (PCs), that is, materials with periodic modulation of their elastic properties. As in the photonic case, the calculation of the effective parameters of a phononic crystal is an important task, for designing of resonant elastic metamaterials and acoustic lenses; there is a great interest in investigating the effective mass density and compliance tensors, as well as the effective sound velocity.

Several homogenization theories, which are valid when the acoustic wavelength is much longer than the lattice constant of the phononic crystal, have been proposed [11–15]. Among homogenization theories, we can identify two commonly used approaches. One of them provides effective acoustic parameters within the framework of multiple scattering [11–13] in which the effective parameters are not obtained according to the direction of propagation; for example, for a sonic crystal the solid material is simply considered rigid; the study is done on the propagation of sound in the air of the structure’s matrix, neglecting the modes that can propagate in the solid. The second, which is based on the Fourier formalism, makes use of the expansion of the microscopic acoustic field into plane waves and allows calculating the effective parameters as a function of the direction of propagation [14, 15]. The latter approach is of particular interest because it can be applied to different geometries of the inclusions inside the unit cell. Here, we will precisely use the Fourier formalism to calculate the effective velocities of elastic wave propagation in 1D solid-solid and fluid-solid phononic crystals. The very first known experimental observation of 1D phononic crystals was when Narayanamurti et al. investigated the propagation of high-frequency phonons through a GaAs/AlGaAs superlattice; other domains in which 1D phononic crystals have potential applications are crystal sensors, waveguide, and resonant transmission [16–19]. As in the case of 1D photonic crystals, such an inherently anisotropic elastic system is a potential metamaterial. However, the main goal of the present work is to establish the basis for the development of a general homogenization theory of (1D, 2D, or 3D) elastic phononic crystals based on the plane wave expansion.

The work is divided as follows: in Section 2, the Fourier formalism applied to a binary elastic superlattice is presented. The proposed method for calculating the effective acoustic parameters for a phononic superlattice, which is based upon the derivation of the wave equation for the macroscopic displacements field, is described in Section 3. Finally, we shall apply the derived explicit formulas for the effective sound velocities to binary superlattices composed of W-Al and Polyethylene-Si and thus study the behavior of this parameter effective in crystals with high and low contrast between the mechanical properties of its components; also a case solid-fluid is presented.

#### 2. One-Dimensional Phononic Crystals

Let us consider a one-dimensional phononic crystal (or elastic superlattice) composed of alternating layers of isotropic elastic materials, A and B (Figure 1). Their thicknesses are, respectively, and , where is the lattice constant. Assuming that the -axis is parallel to the superlattice growth direction, the mass density , as well as the longitudinal and transverse sound velocities, and , turn out to be functions of the -coordinate only.