## Abstract

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

## 1. Introduction

Many studies in the theory of composite materials are based on the exploitation of the classical continuum model implying that the original heterogeneous medium can be simulated by a homogeneous one with certain homogenized (so-called *effective*) mechanical properties. Such an approach comes naturally from the hypothesis of the perfect rate of microheterogeneity of the composite structure when the microscopic size *l* of heterogeneities is supposed to be essentially smaller than the macroscopic size *L* of the whole sample of the material so that in the first approximation one may assume *l*/*L*=0. However, this limit is never reached for most practical problems and in real composites the microstructural scale effects may result in specific non-local phenomena, which cannot be predicted in the frame of the homogenized medium theory.

Scale effects can be systematically analysed by means of the higher order asymptotic homogenization method (AHM). According to this approach, physical fields in a composite are represented by multiple-scale asymptotic expansions in powers of a natural small parameter *ϵ*=*l*/*L*; here *ϵ* characterizes the *rate of heterogeneity* of the structure. This leads to a decomposition of the final solution into macro- and microcomponents, which can be evaluated successively from a recurrent sequence of *cell* boundary-value problems. Further, application of the volume-integral homogenizing operator provides a link from the micro- to macroscopic behaviour of the material and allows the evaluation of effective properties. Theoretical foundations of the method have been developed by Bakhvalov (1974), Babuska (1976), Bensoussan *et al*. (1978), Sánchez-Palencia (1980) and Bakhvalov & Panasenko (1989); a number of recent applications can be found in the works of Meguid & Kalamkarov (1993), Parton & Kudryavtsev (1993), Jikov *et al*. (1994), Kalamkarov & Kolpakov (1997), Boutin *et al*. (1998), Boutin (2000), Rodríguez-Ramos *et al*. (2001), Andrianov *et al*. (2002*a*,*b*, 2005, 2007*a*,*b*), Manevitch *et al*. (2002), Miehe *et al*. (2002), Périn (2004), Berdichevsky (2005), Berger *et al*. (2005), Guinovart-Díaz *et al*. (2005), Kamiński (2005), Parnell & Abrahams (2006) and Santos *et al*. (2006).

Most of the authors quoted above focused on the evaluation of the leading-order terms corresponding to the case of perfectly microheterogeneous structures (*ϵ*=0). The higher order terms arising in static problems were considered by Gambin & Kröner (1989) and Boutin (1996). They have shown that the heterogeneity of the medium results in the induction of an infinite series of displacement fields with successively lower amplitudes. It also causes non-local effects on a macrolevel: instead of the homogenized equilibrium equations of continuum mechanics, we obtain new equilibrium equations that involve the higher order spatial derivatives and thus represent the influence of the microstructural heterogeneity on the macroscopic behaviour of the material. Mathematical aspects of the higher order homogenization have been intensively developed (Smyshlyaev & Cherednichenko 2000; Cherednichenko & Smyshlyaev 2004). Non-local phenomena resulting from a high contrast (or anisotropy) of composite structures were studied by Allaire (1992), Zhikov (2000) and Cherednichenko *et al*. (2006).

In dynamic problems, the physical role of the scale effects is more significant. When the wavelength of a travelling signal decreases and becomes comparable with the characteristic size of heterogeneities, successive reflections and refractions of the local waves at the component interfaces lead to dispersion and attenuation of the global wave field. In order to give a theoretical description of these phenomena, several higher order continuum models were proposed by Achenbach & Herrmann (1968), Sun *et al*. (1968), Bedford & Stern (1971, 1972) and Hegemier *et al*. (1973); for a detailed review of the subject, we refer to Ting (1980). Various self-consistent schemes have been adopted for the purposes of dynamic homogenization by Sabina & Willis (1988), Kanaun & Levin (2003, 2005) and Kanaun *et al*. (2004). The application of the higher order AHM provides a long-wave approach valid in the low-frequency range (Boutin & Auriault 1993; Boutin 1995; Chen & Fish 2001; Fish & Chen 2001, 2004; Bakhvalov & Eglit 2005).

Meantime, a further decrease in the wavelength reveals in a heterogeneous elastic material a complicated structure of the so-called *pass* and *stop* frequency bands. In the literature, they are also referred to as *acoustic* or *phononic* bands (by analogy with *photonic* bands arising for electromagnetic and optical waves in heterogeneous dielectric media). Thus, the composite plays the role of a discrete *wave filter*. If the frequency of the signal falls within a stop band, a stationary wave is excited and neighbouring heterogeneities (e.g. particles) vibrate in alternate directions. On a macrolevel, the amplitude of the global wave is attenuated exponentially, so no propagation is possible.

These general properties are defined for infinite unbounded media; however, they are in very close relation to the *selective reflection* phenomenon shown by bounded composites. A wave striking from outside to a composite medium may be partially reflected and partially transmitted, if the composite is able to transmit the corresponding frequency. Normally, inside the pass bands, the amount of the reflection is defined by a matching problem at the composite boundary. But inside the stop bands, no transmission is allowed and, hence, the wave must be reflected totally.

Experimental observations of the stop bands in heterogeneous elastic structures have been reported by Tamura *et al*. (1991), Martínez-Sala *et al*. (1995), Montero de Espinosa *et al*. (1998), Torres *et al*. (1999), Liu *et al*. (2000), Penciu *et al*. (2000) and Russell *et al*. (2003). An excellent presentation of experimental results and background theory on acoustic wave propagation in periodic solids was given in the monograph by Wolfe (1998).

Obviously, the effects described above can have a great practical importance. Theoretical prediction of the phononic band structures may help to design new composite materials for a large variety of engineering applications, such as a vibrationless environment for high-precision mechanical systems, acoustic filters and noise control devices, ultrasonic transducers, etc.

Another significant property that can be shown by periodically heterogeneous structures is the negative refractive index. This phenomenon is not found for any naturally occurring material. Man-made composites possessing negative refraction are very promising for many optical and microwave applications. In particular, they are used in the construction of *superlenses* providing resolution several times better than the diffraction limit (Pendry 2000; Grbic & Eleftheriades 2004).

In order to explore the high-frequency range, we can represent the unknown solution as an effective wave modulated by some spatially periodic function; such a modulation aims to describe the influence of the composite microstructure. This approach is known as the Floquet–Bloch method, named after the original theorems of Floquet (1883) and Bloch (1928). It has been documented in the classical book of Brillouin (2003) and later used by many authors. Then we come to a spectral eigenvalue problem that allows us to evaluate the structure of bands.

In a one-dimensional case (e.g. for layered composite materials), it is usually possible to derive exact dispersion equations (Silva 1991; Bedford & Drumheller 1994; Ruzzene & Baz 2000; Shul'ga 2003*a*,*b*). In two- and three-dimensional cases (e.g. for fibre-reinforced and grain-reinforced composites), one of the following approaches may be used:

the plane-wave (PW) expansions method (Sigalas & Economou 1992, 1993, 1994, 1996; Kushwaha

*et al*. 1993, 1994; Vasseur*et al*. 1994; Kushwaha & Halevi 1994, 1996; Kushwaha 1997),the Korringa–Kohn–Rostoker method (also known as the multiple-scattering theory, see Kafesaki & Economou (1999), Liu

*et al*. (2000) and Psarobas*et al*. (2000), andthe Rayleigh multipole expansions method and its generalizations (Nicorovici

*et al*. 1995; Movchan*et al*. 1997, 2002; Poulton*et al*. 2000; Platts*et al*. 2002, 2003; Zalipaev*et al*. 2002; Guenneau*et al*. 2003).

All of these methods approximate unknown fields in a heterogeneous medium by some infinite series expansions; they may exhibit convergence difficulties for densely packed high-contrast composites.

McIver (2007) developed the method of matched asymptotic expansions to obtain the dispersion relation and predict the appearance of band gaps for a doubly periodic array of rigid scatterers. High-frequency homogenization using two-scale asymptotic methods in a combination with the Floquet–Bloch approach was proposed by Allaire & Conca (1998) and Cherednichenko & Guenneau (2007). Localization of electromagnetic modes in periodic lattices with defects was explored by Movchan *et al*. (2007).

The scope of the present paper is an application of the higher order AHM to the study of wave propagation in periodic composite materials. Section 2 deals with longitudinal waves in a composite rod (one-dimensional case). The input dynamic problem is formulated in §2*a*. The AHM is applied in §2*b*. Exact solutions of the cell problems up to the order *ϵ*^{3} and the macroscopic wave equation of the order *ϵ*^{2} are obtained in closed analytical forms. Using the Floquet–Bloch approach, the exact dispersion equation is derived and the band structure is analysed in §2*c*. Numerical examples are presented in §2*d*. Section 3 presents the transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional case). Section 3*a* introduces the input dynamic problem. The homogenization procedure is developed in §3*b*; approximate asymptotic solutions of the cell problems up to the order *ϵ*^{3} and the macroscopic wave equation of the order *ϵ*^{2} are obtained. The band structure is evaluated in §3*c* by the PW expansions method. Numerical examples are considered in §3*d*. Finally, §4 presents brief conclusive remarks.

## 2. Longitudinal waves in a composite rod

### (a) Input dynamic problem

Let us consider longitudinal waves propagating in the *x*-direction through a spatially infinite periodic composite rod consisting of alternating layers of two elastic materials *Ω*^{(1)} and *Ω*^{(2)} (figure 1). The governing one-dimensional wave equation is(2.1)where *E*^{(a)} are Young's modulus; *ρ*^{(a)} are mass densities; and *u*^{(a)} are displacements. Here and in the sequel, the superscript (*a*) denotes different components of the composite structure, *a*=1, 2.

Equation (2.1) has to be accompanied by the perfect bonding conditions at the interface *∂Ω*, implying equalities of displacements(2.2)and stresses(2.3)

The dynamic boundary-value problem (BVP) (2.1)–(2.3) allows us to derive the exact dispersion equation (see §2*c*). Thus, it can be considered as a testing example for a practical verification of the higher order AHM. Furthermore, the homogenization procedure developed below may also be extended to the cases when exact solutions do not exist (e.g. multi-dimensional problems, nonlinear composites, etc.).

### (b) Asymptotic homogenization procedure

We start with the analysis of the BVP (2.1)–(2.3) by the AHM (Bakhvalov & Panasenko 1989). Let us define a natural small parameter(2.4)characterizing the rate of heterogeneity of the composite structure. Here the microscopic size *l* corresponds to the length of a periodically repeated unit cell, while the macroscopic size *L* is associated with the wavelength of the travelling signal. In order to separate macro- and microscale components of the solution, we introduce so-called *slow x*- and *fast y*-coordinate variables(2.5)and search the displacements as an asymptotic expansion(2.6)The first term *u*_{0} represents the homogenized part of the solution; it changes slowly within the whole sample of the material and does not depend on the fast coordinate (). The next terms , *i*=1, 2, 3, …, provide corrections of the order *ϵ*^{i} and describe local variations of the displacements on the scale of heterogeneities. The spatial periodicity of the medium induces the same periodicity for with respect to *y*(2.7)The derivatives read(2.8)

Substituting expressions (2.5), (2.6) and (2.8) into the input BVP (2.1)–(2.3) and splitting it with respect to *ϵ*, we come to a recurrent sequence of cell BVPs involving microscopic wave equations(2.9)where *i*=1, 2, 3, …, , and microscopic perfect bonding conditions(2.10)(2.11)

It should be noted that the presented splitting by *ϵ* is formally correct if the physical constants *E*^{(a)}, *ρ*^{(a)} are of the same asymptotic order , . Different orders of the properties of the components (e.g. in the case of extremely high-contrast materials) would change the splitting scheme and may result in non-local effects in the final homogenized equations (Allaire 1992; Zhikov 2000; Cherednichenko *et al*. 2006).

Owing to the periodicity of (2.7), equations (2.9)–(2.11) can be considered within only one periodically repeated unit cell (figure 2) of the composite structure. Bakhvalov & Panasenko (1989) have shown that for a spatially infinite composite with an axially symmetric unit cell the periodicity condition (2.7) may be equivalently replaced by zero boundary conditions (BCs) at the centre and at the outer boundary of the cell(2.12)

Solution of the *i*th BVP (2.9)–(2.12) allows us to evaluate the term . Knowing , we apply the homogenizing operator over the unit cell domain to the (*i*+1)th equation (2.9). Terms are eliminated taking into account that condition (2.11) and periodicity relation (2.7) imply(2.13)As the result, we obtain the *homogenized* wave equation of the order *ϵ*^{i−1}(2.14)Combining together the homogenized equations (2.14) at *i*=1, 2, 3, and so on, we shall obtain macroscopic wave equations of different orders.

Exact analytical solutions of the cell BVPs (2.9)–(2.12) for *i*=1,2,3 are presented in the electronic supplementary material (§S1). The macroscopic wave equation of the order *ϵ*^{2} is as follows:(2.15)where(2.16)and where 〈*E*〉_{0} is the *O*(*ϵ*^{0}) effective Young's modulus; is the homogenized mass density; and *c* is the volume fraction of the component *Ω*^{(2)}, , . Expression (2.16) coincides with the well-known formula for the effective elastic modulus of a layered composite material obtained by simple arithmetical averaging of the components compliances. The parameter(2.17)can be treated as the *O*(*ϵ*^{2}) effective modulus. Formula (2.17) coincides with that obtained by Fish & Chen (2001).

The second term in the l.h.s. of equation (2.15) predicts the effect of dispersion caused by the scattering of the global wave at the local heterogeneities of the composite structure. It can be easily seen that 〈*E*〉_{2} will vanish and the dispersion effect will disappear (i) if the material is homogeneous (*c*=0, 1) and (ii) if acoustic impedances of the components are identical (); the latter case means that there will be no wave reflections at the component interfaces.

Let us consider a harmonic wave(2.18)with the amplitude *U*; the frequency *ω*; and the wavenumber *μ*=2*π*/*L*. Substituting expression (2.18) into the wave equation (2.15), we obtain the dispersion relation(2.19)where , *ω*_{0}=*μv*_{0} is the frequency and is the wave velocity in the quasi-homogeneous (so-called *quasi-static*) limit at *ϵ*=0. The phase *v*_{p} and the group *v*_{g} velocities are as follows:(2.20)(2.21)

The obtained asymptotic solutions (2.19)–(2.21) represent a long-wave approximation supposing that the wavelength *L* is considerably larger than the size *l* of heterogeneities of the composite structure. As *L* decreases (the frequency *ω* increases), the group velocity *v*_{g} decreases. The condition *v*_{g}=0 determines the end of the first pass band.

The developed homogenization procedure is formally valid for spatially infinite media. In the case of bounded composites the higher order microscopic solutions for should depend on the global boundary conditions (BCs) imposed on the structure at macrolevel. Direct application of the higher order solutions, developed for spatially infinite media, to bounded structures can lead to errors in the global BCs.

### (c) Floquet–Bloch approach and phononic bands

In order to explore the high-frequency range and to develop a solution valid for short waves, we study the input wave equation (2.1) by the Floquet–Bloch approach (Silva 1991; Bedford & Drumheller 1994; Ruzzene & Baz 2000; Brillouin 2003). According to this method, a harmonic wave propagating through a periodic composite material can be represented in the form(2.22)where is a spatially periodic function aiming to describe the influence of the composite microstructure, . The periodicity of yields(2.23)

Substituting expression (2.21) into the BVP (2.1)–(2.3) and taking into account the quasi-periodicity relation (2.22), we shall obtain a system of four linear algebraic equations for the unknown coefficients , . Details of the evaluations can be found in the electronic supplementary material (§S2). Such a system has a non-trivial solution if and only if the determinant of the matrix of the coefficients is zero. Equating the determinant to zero, we derive the exact dispersion equation for *ω* and *μ*(2.24)Equation (2.24) was obtained by many authors (Vladimirskii 1946; Krein & Liubarskii 1961; Pobedrya 1984; Silva 1991; Bedford & Drumheller 1994; Ruzzene & Baz 2000; Shul'ga 2003*b*). It represents a solution for steady-state elastic waves in one-dimensional periodically heterogeneous media.

In the long-wave approximation, settingwe can determine *ω* as an asymptotic series in powers of *ϵ*. Coefficients of such a series, computed from equation (2.24), shall coincide with the leading terms of expansion (2.19) obtained by the higher order AHM.

In order to reveal the appearance of phononic band gaps, we rewrite expression (2.22), separating real *μ*_{R} and imaginary *μ*_{I} parts of the wavenumber *μ*=*μ*_{R}+i*μ*_{I},(2.25)It can be easily seen that if the wave propagates at a frequency making the wavenumber complex then the signal (2.25) attenuates exponentially; here *μ*_{I} represents the attenuation factor. Thus, the frequency bands where *μ*_{I}≠0 are called stop bands, while the bands where *μ*_{I}=0 are called pass bands. Qualitative analysis of the band structure is presented in the electronic supplementary material (§S3).

### (d) Numerical examples

As the first example, we consider a low-contrast composite consisting of aluminium (*E*^{(1)}=70 GPa, *ρ*^{(1)}=2700 kg m^{−3}) and steel (*E*^{(2)}=210 GPa, *ρ*^{(2)}=7800 kg m^{−3}). A general structure of the pass and stop bands, depending on the steel volume fraction *c*, is presented in the electronic supplementary material in figure S3. Figure S4 displays the phononic bands evaluated by equation (2.24) at *c*=0.3. Figure S5 compares the dispersion curve in the first pass band (acoustic branch) calculated by the AHM (formula (2.19)) with the numerical solution of the exact dispersion equation (2.24). We can observe that the AHM allows us to predict the dispersion effect, but the numerical accuracy of the obtained results is acceptable only at low frequencies.

As the second example, let us consider a high-contrast composite consisting of alternating layers of carbon/epoxy lamina (*E*^{(1)}=8.96 GPa, *ρ*^{(1)}=1600 kg m^{−3}) and steel (*E*^{(2)}=210 GPa, *ρ*^{(2)}=7800 kg m^{−3}). The structure of the pass and stop bands is shown in figure S6. Phononic bands evaluated by equation (2.24) at *c*=0.5 are presented in figure S7. For the acoustic branch, the AHM shows a good agreement with the exact dispersion equation; this is shown in figure S8. Figures S3–S8 are given in the electronic supplementary material.

## 3. Anti-plane shear waves in a square lattice of cylinders

### (a) Input dynamic problem

We study transverse anti-plane shear waves propagating in the *x*_{1}*x*_{2} plane through a fibre-reinforced composite material consisting of an infinite matrix *Ω*^{(1)} and a periodic square lattice of cylindrical inclusions *Ω*^{(2)} (figure 3). The governing two-dimensional wave equation is as follows:(3.1)where *G* is the shear modulus; *ρ* is the mass density; *w* is the longitudinal displacement (in the *x*_{3}-direction); and , *e*_{1}, *e*_{2} are the unit Cartesian vectors. Owing to the heterogeneity of the medium, the physical properties *G* and *ρ* are represented by piecewise continuous functions of coordinates(3.2)where . Equation (3.1) can be written in the equivalent form(3.3)where *G*^{(a)}, *ρ*^{(a)} are the properties of the components, .

The perfect bonding conditions at the components interface *∂Ω* correspond to the equalities of displacements(3.4)and tangential stresses(3.5)where *∂*/*∂*** n** is the normal derivative to

*∂Ω*.

It should be noted that the scalar BVP (3.3)–(3.5) can possess different physical interpretations. Here we shall discuss it with reference to the anti-plane shear. However, it is mathematically identical to problems of in-plane propagation of electromagnetic and optical waves through an array of dielectric cylinders.

### (b) Asymptotic homogenization procedure

Let us analyse the BVP (3.3)–(3.5) by the AHM (Bakhvalov & Panasenko 1989). We follow the algorithm described in §2*b*. The natural small parameter *ϵ* is defined by equation (2.4). Slow *x*_{s}- and fast *y*_{s}-coordinates are as follows:(3.6)The displacement field is searched as an asymptotic expansion(3.7)where . The homogenized term *w*_{0} does not depend on fast coordinates, . The higher order terms , *i*=1,2,3, …, are *y*-periodic due to the spatial periodicity of the medium(3.8)where , , , *l*_{s} are the fundamental translation vectors of the square lattice, *l*_{s}=*l**e*_{s}. The differential operators read(3.9)where , , .

Let us assume that the physical properties of the components are of the same asymptotic order, , . Splitting the input BVP (3.3)–(3.5) with respect to *ϵ* leads to a recurrent sequence of cell BVPs involving microscopic wave equations(3.10)where *i*=1,2,3, …, , and microscopic perfect bonding conditions(3.11)(3.12)where *∂*/*∂*** m** is the normal derivative to the interface

*∂Ω*written in fast variables. The periodicity of (3.8) implies that equations (3.10)–(3.12) can be considered within a distinguished periodically repeated unit cell (figure 4) of the composite structure. Following Bakhvalov & Panasenko (1989), we replace the periodicity equation (3.8) by zero BCs at the centre and at the outer boundary

*∂Ω*

_{0}of the unit cell(3.13)

Solution of the *i*th BVP (3.10)–(3.13) determines the term . Then the homogenizing operator over the unit cell domain should be applied to the (*i*+1)th equation (3.10). Terms are eliminated by means of Green's theorem, which together with the boundary condition (3.12) and the periodicity relation (3.8) implies(3.14)As the result, the homogenized wave equation of the order *ϵ*^{i−1} shall be obtained(3.15)Combination of the homogenized equations (3.15) at *i*=1,2,3, … provides macroscopic wave equations of different orders.

Unlike the one-dimensional example considered in §2, the cell BVP (3.10)–(3.13) do not allow exact analytical solutions. Therefore, some approximate approaches should be applied. Usually, the main computational difficulties arise in the case of high-contrast densely packed composites. Interactions between neighbouring inclusions induce rapid oscillations of the stress field on a microlevel. As the volume fraction of the inclusions and the contrast between the components properties increase, the local gradients can grow significantly. Then many of the commonly used methods may lack convergence: analytical approaches that represent physical fields by various series expansions need to calculate a number of additional terms of the series; the finite-element method requires a drastic increase in the density of the discretization mesh, etc. In the present paper, we focus on the most difficult case of a high-contrast densely packed composite and propose an asymptotic solution of the cell problems using as a natural small parameter the non-dimensional distance *δ*=2*H*/*L* between the neighbouring inclusions (figure 4).

Let us suppose *δ*≪1 and *G*^{(2)}/*G*^{(1)}≫1. Being restricted by the *O*(*δ*^{0}) approximation, for the matrix strips d*Ω*_{1}, d*Ω*_{2} that separate neighbouring inclusions, we can easily show (Andrianov *et al*. 2002*a*,*b*)(3.16)The physical meaning of estimations (3.16) is that in the narrow strip d*Ω*_{1} the variation of local stresses in the *y*_{1}-direction is dominant and, hence, the term can be neglected in comparison with . The other way around, in the strip d*Ω*_{2}, the dominant variation of the local stress field takes place in the *y*_{2}-direction, so the term can be neglected in comparison with . Such a simplification is similar to the basic idea of the well-known *lubrication theory*, which was used in the theory of composites for many years (Frankel & Acrivos 1967; Christensen 1979).

Following estimations (3.16), in the *O*(*δ*^{0}) approximation equation (3.10) reads:(3.17)Solution of the simplified BVP (3.11)–(3.13), (3.17) can now be derived in a closed analytical form (see §S5 in the electronic supplementary material).

The macroscopic wave equation of the order *ϵ*^{2} is as follows: (3.18)where 〈*G*〉_{0}, 〈*G*〉_{2} are, respectively, the *O*(*ϵ*^{0}) and the *O*(*ϵ*^{2}) effective shear modulus; *c* is the volume fraction of the inclusions *Ω*^{(2)}, *c*=*πA*^{2}/*L*^{2}; and *A* is the radius of the inclusion (figure 4), , , . The moduli 〈*G*〉_{0}, 〈*G*〉_{2} are determined by the evaluation of the integrals in equation (3.15). This was performed numerically in the program package Maple using standard in-built subroutines.

In order to verify the developed solution, in figure S9, we compare the asymptotic behaviour of 〈*G*〉_{0} with an asymptotic formula of O'Brien (1977) in the case of high-contrast () densely packed (*c*→*c*_{max}) composites. It should be noted that the obtained solution shows a practically acceptable accuracy at all values of the volume fractions and properties of the components. This is shown in figure S10, where our results for 〈*G*〉_{0} are compared with convergent-proved data of Perrins *et al*. (1979).

Let us consider a harmonic wave(3.19)where *W* is the amplitude; *ω* is the frequency; is the wavevector; and is the wavenumber, , , .

Substituting expression (3.19) into the wave equation (3.18), we obtain the dispersion relation as follows:(3.20)where , is the frequency and is the wave velocity in the quasi-static case. The phase *v*_{p} and the group *v*_{g} velocities are as follows:(3.21)(3.22)The asymptotic solutions (3.20)–(3.22) represent a long-wave approximation. The upper frequency limit *ω*_{max} of the first pass band can be determined from the condition *v*_{g}=0.

In the quasi-static case, the composite under consideration is transversely orthotropic. The *O*(*ϵ*^{0}) solution for anti-plane shear is isotropic in the transverse plane *x*_{1}*x*_{2}, so the parameters 〈*G*〉_{0}, *v*_{0}, *ω*_{0} do not depend on the direction of the wave propagation. The higher order approximations reveal the anisotropy of the problem; the *O*(*ϵ*^{2}) solutions (3.20)–(3.22) depend explicitly on the direction *φ* of the wavevector. The effect of anisotropy can also be observed at numerical examples presented in §3*d*.

### (c) PW expansions method

In order to develop a solution valid in the high-frequency range, let us study the input wave equation (3.1) by the PW expansions method (Sigalas & Economou 1992; Kushwaha *et al*. 1994). Following the Floquet–Bloch theorem (Brillouin 2003), we represent a harmonic wave propagating through a periodic composite material as follows:(3.23)where *F*(** x**) is a spatially periodic function, .

Let us expand the function *F*(** x**) and the material properties

*G*(

**),**

*x**ρ*(

**) into infinite Fourier series as follows:(3.24)wherethe operator denotes integration over a distinguished unit cell**

*x**Ω*

_{0}.

Substituting Ansatz (3.23) and expansions (3.24) into the wave equation (3.1) and collecting the terms , , we come to an infinite system of linear algebraic equations for the unknown coefficients as follows:(3.25)System (3.25) has a non-trivial solution if and only if the determinant of the matrix of the coefficients is zero. Equating the determinant to zero, we derive a dispersion relation for *ω* and *μ*. It should be noted that the PW expansions method does not use explicitly the bonding conditions (3.4) and (3.5), whereas they are ‘embedded’ implicitly into equation (3.1) and expansions (3.24).

In the numerical examples presented below, the dispersion relations are calculated approximately by the truncation of the infinite system (3.25), supposing . The number of the kept equations is . We expect that increase in *j*_{max} shall improve the accuracy of the solution. From the physical point of view, such a truncation means cutting off the higher frequencies.

A theoretical shortcoming of the PW expansions method consists in the Gibbs phenomenon. As the functions *F*(** x**),

*G*(

**) and**

*x**ρ*(

**) undergo jump discontinuities at the interface**

*x**∂Ω*, truncated Fourier series (3.24) displays local overshoots (so-called

*ringing artefacts*). If more terms are added (i.e.

*j*

_{max}increases), the width of the overshoots becomes smaller, but their amplitude remains about the same (roughly 9%).

To illustrate the appearance of phononic band gaps, let us rewrite Ansatz (3.23) separating real *μ*_{R} and imaginary *μ*_{I} parts of the wavevector as follows:(3.26)The imaginary part of the wavenumber represents the attenuation factor. Frequency regions where *μ*_{I}≠0 correspond to stop bands (signal (3.26) attenuates exponentially), while regions where *μ*_{I}=0 correspond to pass bands.

### (d) Numerical examples

We begin with a comparison of the PW expansions method with a solution of Poulton *et al*. (2000) obtained by the Rayleigh method. Let us consider a square array of circular cavities (*G*^{(2)}=0, *ρ*^{(2)}=0) embedded in a matrix with normalized properties *G*^{(1)}=1, *ρ*^{(1)}=1. Figure S11*a*,*b* shows dispersion curves for the cavities of non-dimensional radius (*c*≈0.126) and *r*_{c}=0.4 (*c*≈0.503). Each dispersion diagram consists of two (left and right) parts separated by a vertical dashed line. The right part displays a solution for the orthogonal direction (*φ*=0) and the left part for the diagonal direction (*φ*=*π*/4) of the wave propagation. We can observe that in the quasi-static case (*ω*→0) the solution is isotropic; however, with the increase in *ω* the composite structure exhibits an anisotropic behaviour. Shaded areas indicate the so-called *full* stop bands, where anti-plane shear waves are damped in any direction. Increase in *j*_{max} improves the accuracy of the solution; at *j*_{max}=2 the presented results almost coincide with the data of Poulton *et al*. (2000).

Next, let us consider a low-contrast composite consisting of nickel fibres (*G*^{(2)}=75.4 GPa, *ρ*^{(2)}=8936 kg m^{−3}, *c*=0.35) and an aluminium matrix (*G*^{(1)}=27.9 GPa, *ρ*^{(1)}=2700 kg m^{−3}). Figure S12a displays dispersion curves within the first and second pass bands calculated by the PW expansions method. Results obtained at *j*_{max}=1 and 2 are very close, which confirm a fast convergence of the procedure. Figure S12b compares the acoustic branches calculated by the AHM and the PW expansions method. Qualitatively, the AHM predicts the effect of dispersion, but its numerical accuracy is acceptable only at low frequencies.

As an example of a high-contrast composite, we consider carbon fibres (*G*^{(2)}=86 GPa, *ρ*^{(2)}=1800 kg m^{−3}, *c*=0.5) embedded in an epoxy matrix (*G*^{(1)}=1.53 GPa, *ρ*^{(1)}=1250 kg m^{−3}). Figure S13 displays the acoustic branch of the dispersion curve for orthogonal waves (*φ*=0). It should be noted that the PW expansions method converges much slowly than in the low-contrast case and large computational efforts are required to achieve a good accuracy even for small frequencies. Meanwhile, the AHM provides qualitatively correct results up to the beginning of the first stop band (i.e. at *μ*_{R}*l*=*π* the AHM gives *v*_{g}≈0).

## 4. Conclusions

In the present paper, longitudinal waves in a composite rod and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions are studied. Successive reflections and refractions of the signal at the component interfaces result in the formation of pass and stop frequency bands, so the composite acts as a discrete wave filter. Application of the higher order AHM provides a long-wave approach valid in the low-frequency range. Solution for the high frequencies is obtained by the PW expansions method. However, this approach may run into convergence problems with the increase in contrast between the components' properties. Eventually, we may conclude that the higher order AHM and the PW expansions method can be treated as complementary to each other. As a result, the dispersion curves are obtained, the pass and stop frequency bands are identified.

## Acknowledgments

This work was supported by the Alexander von Humboldt Foundation (Institutional academic cooperation programme, grant no. 3.4-Fokoop-UKR/1070297) and by the German Research Foundation (Deutsche Forschungsgemeinschaft, grant no. WE 736/25-1). The authors are grateful to anonymous reviewers for their valuable comments and suggestions that helped to improve the paper.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0267 or via http://journals.royalsociety.org.

- Received October 12, 2007.
- Accepted January 14, 2008.

- © 2008 The Royal Society