We investigate the use of acoustic metamaterials to design structural materials with frequency selective characteristics. By exploiting the properties of acoustic metamaterials, we tailor the propagation characteristics of the host structure to effectively filter the constitutive harmonics of an incoming broadband excitation. The design approach exploits the characteristics of acoustic waveguides coupled by cavity modes. By properly designing the cavity we can tune the corresponding resonant mode and, therefore, coupling the waveguide at a prescribed frequency. This structural design can open new directions to develop broadband passive vibrations and noise control systems fully integrated in structural components.

## I. INTRODUCTION

Over the past decade, acoustic metamaterials have attracted a growing interest due to their ability to modify an incoming acoustic wave field in ways not attainable using conventional materials. A typical arrangement for solid elastic metamaterials is represented by a background material having embedded and periodically distributed cylindrical inclusions made out of a different material. The inclusions can also be made out of two or more materials, typically leading to what is referred to as resonant acoustic metamaterial.^{1,2} The performance of the metamaterial in tailoring propagating wave fronts is strictly connected to the nature of the constitutive materials and the resulting mechanical impedance mismatch at their interface.

To-date, the majority of the studies has concentrated on the design and analysis of the performance of metamaterials with particular attention to understanding their peculiar characteristics such as the frequency bandgap,^{1} the negative refraction index,^{3} and the sub-wavelength resolution.^{4–6} The discovery and understanding of these properties have paved the way to a series of potential applications that were not previously conceivable based on the use of conventional materials. Cloaking acoustic devices^{7} are a relevant example of application that have greatly benefited from this new class of engineered materials. In this paper, we explore the possible application of a metamaterial based design to the synthesis of structural materials having frequency selective characteristics. This design can have critical implications for applications on energy harvesting, and passive vibration and noise control systems paving the way to the development of multi-functional structures with passive embedded acoustic filtering capabilities. To design embedded acoustic filters we exploit a well-known property of acoustic metamaterials, the frequency bandgap, combined with a network of waveguides coupled by defect modes. To illustrate the working principle of the actual system we briefly review the characteristics of the different elements separately.

## II. BANDGAPS AND DEFECT MODES

A perfectly periodic metamaterial is characterized by the existence of acoustic bandgaps in its frequency spectrum that identify frequency ranges where propagating waves are not supported. The bandgap effect is related to the back scattering of prescribed wavelengths due to the periodic inclusions. If the metamaterial is not perfectly periodic, due to the existence of an irregularity, localized modes can occur inside the bandgap.^{8} The irregularity can be introduced by any variation of either the geometric or the material properties. This process is similar, in principle, to the localization effect occurring in mistuned periodic structures.^{9,10} To exemplify this mechanism we consider a bulk material with carbon cylinders in a square lattice arrangement embedded in an epoxy matrix. The combination carbon/epoxy was chosen because it is a well suited material for structural applications and because of its large impedance mismatch which results in wide frequency bandgaps. The band structure of the crystal is calculated by using the Plane Wave Expansion method (PWE)^{11,12} applied to a 5 × 5 supercell model.^{9,13} The response of the bulk material is governed by the Navier's governing equations:

where ρ(*x*, *y*, *z*) and *C*_{ijkl}(*x*, *y*, *z*) are the position-dependent mass density and elastic stiffness tensor, respectively. Due to the spatial periodicity of the crystal in the x-y plane and the homogeneity in the z direction, the material constants in eqn. (1), ρ(*x*, *y*, *z*) and *C*_{ijkl}(*x*, *y*, *z*) can be expanded in Fourier series with respect to the two-dimensional reciprocal lattice vectors (RLV), *G* = (*G*_{1}, *G*_{2}) = (*N*_{1}, *N*_{2}) × π/*a* as:

where ρ_{G} and |$C_G^{ijkl}$|$CGijkl$ are the corresponding Fourier coefficients.^{14}

To simulate an infinite bulk material, we use the Bloch theorem and expand the displacement vector *u*(*x, y, z, t*) in Fourier series as follows:

where *k* = (*k*_{x}, *k*_{y}) is the Bloch wave vector, *ω* is the circular frequency and |$A_{G^\prime }$|$AG\u2032$ is the amplitude of the displacement vector. Substituting eqns. (2) and (3) into eqns. (1), and after collecting the terms *e*^{i[k•(x, y) − ωt]}, we obtain the 3n × 3n set of equations:

where the n × n sub-matrices |$C_{G,G^\prime }^{( 1 )},$|$CG,G\u2032(1),$|$C_{G,G^\prime}^{( 2 )}$|$CG,G\u2032(2)$, |$C_{G,G^\prime }^{( 3)}$|$CG,G\u2032(3)$, |$W_{G,G^\prime}^{( 1 )}$|$WG,G\u2032(1)$ and |$W_{G,G^\prime }^{( 2 )}$|$WG,G\u2032(2)$ are functions of the Bloch wave vector *k*, the two-dimensional reciprocal lattice vectors *G*, the circular frequency *ω*, the Fourier coefficients of mass density ρ_{G} and the elastic stiffness tensor |$C_G^{ijkl}$|$CGijkl$.^{14}

Figure 1(a) shows the schematic of the supercell carbon-epoxy metamaterial with a single point defect, represented by a missing inclusion, located in the center. The material properties used in the calculation are as follows: *C*_{11} = 30.96E10 Pa, *C*_{12} = 13.26E10 Pa and *C*_{44} = 8.84E10 Pa for carbon; *C*_{11} = 0.96E10 Pa, *C*_{12} = 0.64E10 Pa and *C*_{44} = 0.16E10 Pa for epoxy; the density for carbon is *ρ* = 1750 kg/m^{3} and for epoxy is *ρ* = 1200 kg/m^{3}. The filling fraction is *f* = *πr*^{2}/*a*^{2} = 0.55. By comparing the band structure of the perfect metamaterial crystal (Figure 1(c)) with that of the defected crystal (Figure 1(d)) we note that the presence of the cavity induces four localized modes located inside the bandgap (red lines in Figure 1(d)). As an example, the modal displacement of the third defect mode (Ω = ω × a/2π*C*_{t} = 0.1888, where *C*_{t} = |$\sqrt {\frac{{C44}}{\rho }}$|$C44\rho $ is the transverse wave velocity for carbon) is also shown in Figure 1(b). Note that the bulk wave is decoupled into two sets: in-plane (Longitudinal (L) and Shear Horizontal (SH)) modes where the particle vibration is in the x-y plane and the Shear Vertical (SV) modes where the particle vibration is along the z direction (see the coordinates system in Figure 1(a)). Here we only consider the SV mode to investigate the coupling effects of the guided structure, which means that only the elastic coefficient *C*_{44} and the density *ρ* are needed to calculate the corresponding band structure.

## III. COUPLING MECHANISM

This mechanism is of particular interest because it allows us to couple different areas of the crystal at prescribed frequency ranges. Similarly to what observed for a point defect, the introduction of a waveguide (also called line defect) in a perfectly periodic metamaterial crystal (see Figure 2(a) or 2(b), channel #1) results in additional localized modes corresponding to dispersion curves branches located inside the bandgap. These branches identify modes that can freely propagate inside the waveguide. We introduce a second waveguide (see Figure 2(a) or 2(b), channel #2) geometrically identical to the first and separated from it by a periodic crystal with finite size (indicated by the dashed red box in Figure 2(a) and 2(b)). The two waveguides will exhibit identical modes but their dynamic response will be decoupled by the presence of the bandgap associated with the interposed perfectly periodic crystal. This concept is exemplified in Figure 2(c) where the band structures of the two channels are compared with that of a perfectly periodic crystal (Figure 2(c), center, black curves).

The band structure of the single waveguide is extracted by applying the PWE method to a 5 × 5 supercell. The supercell is obtained by removing an entire row of inclusions from the initially perfectly periodic crystal, as indicated in Figure 2(a) (see the blue dashed box). In the absence of a point defect, the bandgap due to the intermediate perfectly periodic crystal decouples the two waveguides preventing energy exchange at any of the frequencies inside the bandgap. Conversely, when a point defect is introduced (Figure 2(b)), the generation of localized defect modes inside the bandgap (red curves in Figure 2 center) will allow wave propagation between the channels at selected frequencies. Note that the previously described situation of interaction between waveguides would be achieved as a limiting case when the interposed crystal size approaches infinity. In fact, due to the finite size of the periodic crystal interposed between the two channels (red dashed box in Figure 2(a) and 2(b)) the inclusions will produce only partial back scattering. As a result, the waveguides will not be fully decoupled and a small percentage of the vibrational energy will still be exchanged between the channels.

Note that, the geometric or material characteristics of the defect can be selected in order to tailor the transfer function between the two channels and, therefore, allowing the exchange of energy only at specific frequencies. That is, the point defect acts as a mechanical bandpass or notch filter with respect to an incoming broadband elastic wave.

## IV. FILTER DESIGN

The concept of coupled waveguides can be exploited to create structures that can effectively filter, separate and recombine the constitutive harmonics of an incoming broadband signal. This approach can enable the fabrication of structural elements with embedded filtering, multiplexing, and demultiplexing functionalities. As an example, we show how this design approach can be exploited to create a structural material that can also act as an acoustic spectrometer. We arrange the embedded inclusions to form a network of six waveguides (Figure 4(a)). The waveguides are created by removing selected rows of elements from the perfectly periodic bulk crystal described above.

The five lateral channels (*drop channels*^{15–18}) are connected to a central waveguide (*main waveguide*) that delivers the incoming broadband signal. The five drop channels are dynamically coupled to the main guide by exploiting the concept of point defect modes. Five different cavities are designed in order to obtain five well separated cavity modes inside the frequency bandgap of the bulk crystal. The frequencies of the cavities are tuned acting on the radius of the inclusions. The dispersion characteristics of the full cavity (r_{d} = 0, i.e. no inclusion) were shown in Figure 1(d). The band structure of the remaining cavities is provided in Figure 3. The final design parameters, including radii and tuning frequencies, are listed in Figure 4(b).

To verify the performance of the spectrometer we performed a time response analysis solving eqn. (1) by using the Finite Difference Time Domain (FDTD) method.^{19,20} Perfectly matched layers^{21} were also used on the crystal boundary to reduce the effect of reflected waves. The main waveguide is excited by a multi-harmonic transverse wave produced by a point force (red dot Figure 4(a)). The excitation force is given by the superposition of five harmonic signals tuned at the frequencies of the five drop channels. The spectral content of the response in the five channels (Figure 4(b)) is obtained by Fourier transforming the time history of the displacement at selected points (light blue dots in Figure 4(a)). Numerical results clearly show that the proposed design can effectively operate as an acoustic spectrometer separating the constitutive harmonics of the incoming signals and redirecting them in selected waveguides. Note that the design could be optimized in order to reduce some of the leakage in channels #1 and #3. The filtering capability of the proposed design is further illustrated by visualizing the induced displacement field in two selected drop channels (Figure 5). In this case, a single tone harmonic excitation is injected in the main guide to show how the different drop channels can be activated by simply selecting the excitation frequency. The performance of each drop channel, in terms of passband width and transmitted power, depends on the size and location of the cavity as well as on the sequence of inclusions in the adjoining section between two channels. Further studies will be performed to explore this dependence. It should be pointed out that, even though we used SV modes to investigate the coupled response of the drop channels, the same concept could be extended to the use of in-plane modes. Nevertheless, in this last case mode conversion should be expected upon reflection of the waves on either geometric or material interfaces. Mode conversion should be carefully considered in the design of the filter because could result in crosstalk effects between the different channels.

## V. CONCLUSIONS

In this paper, we presented a metamaterial based design approach to tailor the wave propagation characteristics of structural materials. Numerical results have shown that this design is particularly effective in creating passive frequency selective structures. Although the concept is exemplified on a bulk material with simple waveguide geometry, this design approach is very general and could be extended to create any complex network of mechanical filters resulting from the combination of bandpass and notch filters. Also, it is envisioned that each drop channel could be coupled with a network of sub-channels, therefore extending this approach to the design of structural materials with embedded multiplexer or demultiplexer capabilities. Due to the basic properties of metamaterials, this design is also expected to be highly scalable and usable for a large spectrum of engineering applications.