**1. Introduction**

In the last two decades, the propagation of elastic waves in periodic composite materials, known as phononic crystals, has attracted increased attention due to their unique physical properties. These properties include phononic bandgaps which define a frequency range where elastic wave propagation is forbidden [1]. The existence of phononic bandgaps enables a variety of potential applications, such as noise and vibration insulation. Two mechanisms cause the formation of bandgaps: Bragg scattering [1] and local resonance [2]. In Bragg scattering, the associated wavelength with the phononic bandgap is of the same order as the periodicity of the structure. This means a huge lattice constant is needed to obtain phononic bandgaps for the low frequency range, which limits applications. To achieve local resonance, the associated wavelength needs to be two orders of magnitude smaller than the Bragg bandgap. A locally resonant bandgap is related to the resonance frequency associated with scattering units and depends less on the periodicity and symmetry of the structure. Therefore, it overcomes the limitation of Bragg bandgaps and permits bandgaps suitable for low frequencies.

The existence of low-frequency locally resonant bandgaps (LRBGs) gives rise to the application of locally resonant phononic crystals (LRPCs) in the reduction of low-frequency vibration and noise [3]. However, it has been a challenging task due to the long wavelength and weak attenuation of the low-frequency waves. In the past years, many LRPCs were proposed to obtain bandgaps in low frequency range or broaden bandgaps in the low frequency range [4–19]. More recently, the propagation of Lamb waves (elastic waves) in phononic crystal plates, which is based on a local resonance mechanism, has attracted attention due to their potential applications in filters, resonators, waveguides, and for vibration insulation. In general, phononiccrystal plates can be classified into two types according to their structural features: flat plates and stubbed plates. The flat phononic-crystal plate consists of a periodical array of holes or periodic inclusions of foreign material in a homogeneous plate [5]. Many previous studies focused on this type of phononic-crystal plates to investigate the bandgap properties. It is not easy to obtain lower frequency phononic bandgaps using a flat phononic-crystal plate. Stubbed phononic-crystal plates can be classified into two types: the single-sided stubbed phononic-crystal plate and the doublesided stubbed phononic-crystal plate. The single-sided stubbed phononic-crystal plate [3] consists of a square array of stubs on one side of a homogeneous plate [6]. Subsequently, many researchers investigated the effects of material properties and geometric parameters on the BG of the single-sided stubbed phononic-crystal plate and found that the mass and the geometric parameter effect can help tune the bandgap into low frequencies [8]. Later, a novel single-sided stubbed phononiccrystal plate was proposed. It consists of a square array of stubs on one side of a two-dimensional binary locally resonant phononic plate. It can increase of the relative bandwidth over the classical single-sided stubbed plate [12]. The double-sided stubbed phononic-crystal plate consists of a square array of stubs on both sides of a homogeneous plate. It can reduce the bandgap compared to a single-sided stubbed phononic-crystal plate [9]. Recently, a novel double-sided stubbed phononic-crystal plate was proposed. Unlike the classical double-sided stubbed phononic-crystal plate, it can shift the bandgap into the lower frequency range [13].

However, because there are three different modes for a plate, and they can only be coupled separately to special resonating modes of the resonators, the gaps for in-plane and out-of-plane plate modes can hardly be overlapped with each other in lower frequencies due to the coupling of the spring-mass system of the resonator, and thus the lower complete bandgaps (below 100 Hz) are difficult to obtain. As a result, the BGs of these disused PC plates are usually located in the frequency range above 300 Hz. However, the frequency of most of the ambient vibration in practical cases is distributed over a wide frequency range from 20 to 250 Hz. Also, the broad bandgaps for in-plane and out-of-plane plate modes can hardly be overlapped with each other in same frequency range due to the weak coupling between the resonator mode and the plate mode, and thus the broad complete bandgaps are difficult to obtain too. At the same time, a common feature of phononic-crystal plates is matrix structures made of nonmetallic materials. However, most mechanical structures typically consist of metals, and a phononiccrystal plate with a nonmetallic material matrix can hardly be used for vibration and noise reduction in mechanical engineering. Recently, a metal-matrix embedded phononic crystal which consists of periodic double-sided stepped resonators deposited on a two-dimensional phononic plate within a steel matrix was proposed. It is found that the bandwidth is increased, but the opening location of the bandgap is higher (1000 Hz) [19]. In other words, the extension of the bandwidth into the lower frequency range remains a challenging task for a metal-matrix embedded phononic crystal.

**31**

**Figure 1.**

*and the high-symmetry M, Γ, and X.*

*Metal-Matrix Embedded Phononic Crystals DOI: http://dx.doi.org/10.5772/intechopen.80790*

**2. The method of calculation**

in a practical case.

in solids are given by

∑

ment field can be expressed as

*j*=1 <sup>3</sup> \_\_\_<sup>∂</sup> <sup>∂</sup>*xj*(<sup>∑</sup> *l*=1 3 ∑ *k*=1 3

*u*(*r*) = *e<sup>i</sup>*(*k*⋅*r*)

between the unit cell and its two adjacent cells, given by

Thus, how to adjust the in-plane and out-of-plane gaps overlapping with each other in the lower frequencies (below 100 Hz) or increasing the in-plane and outof-plane gaps in the same range is an important issue for the control of the vibration

In this chapter, two kinds of metal-matrix embedded phononic crystals are introduced. The one is called the lower frequency complete bandgap metal-matrix embedded phononic crystals [14] because it can produce lower frequency complete bandgap, and the other is called the broad complete bandgap metal-matrix embedded phononic crystals [20] because it can produce broad complete bandgap.

In order to investigate the band properties of the metal-matrix embedded phononic crystals, a series of calculations on dispersion relations and transmission spectra are conducted with FEM based on the Bloch theorem. For the calculation of the dispersion relations, the governing field equations for elastic wave propagation

> *cijkl* \_\_\_\_ ∂*uk* <sup>∂</sup>*xl*) <sup>=</sup> *<sup>ρ</sup>*

where ρ is the mass density, t is the time, ui is the displacement, cijkl are the elastic constants, and xj (j = 1, 2, 3) represents the coordinate variables x, y, and z, respectively. According to the Bloch theorem, in the FEM formulation, the displace-

where u is the displacement at the nodes and r is the position vector located at the boundary nodes. The Bloch wave vector k = (kx, ky) is the wave vector limited to the first Brillouin zone of the reciprocal lattice. Since the metal-matrix embedded phononic crystal plate is periodic in the xy-direction and finite in the z-direction, only the unit cell (shown as in **Figure 1(b)** or **Figure 3(b)**) needs to be considered in the FEM calculation when the Bloch theorem is adopted on the boundaries

*ui*(*x* + *a*, *y* + *a*) = *e<sup>i</sup>*(*kx*⋅*a*+*ky*⋅*a*) *ui*(*x*, *y*)(*i* = *x*, *y*, *z*) (3)

*(a and b) Schematic of the part and the unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals, respectively. (c) The corresponding first irreducible Brillouin zone (red region)* 

∂2 \_\_\_\_*ui* ∂ *t*

<sup>2</sup> (*i* = 1, 2, 3) (1)

*uk*(*r*) (2)

*Metal-Matrix Embedded Phononic Crystals DOI: http://dx.doi.org/10.5772/intechopen.80790*

*Photonic Crystals - A Glimpse of the Current Research Trends*

plate, it can shift the bandgap into the lower frequency range [13].

However, because there are three different modes for a plate, and they can only be coupled separately to special resonating modes of the resonators, the gaps for in-plane and out-of-plane plate modes can hardly be overlapped with each other in lower frequencies due to the coupling of the spring-mass system of the resonator, and thus the lower complete bandgaps (below 100 Hz) are difficult to obtain. As a result, the BGs of these disused PC plates are usually located in the frequency range above 300 Hz. However, the frequency of most of the ambient vibration in practical cases is distributed over a wide frequency range from 20 to 250 Hz. Also, the broad bandgaps for in-plane and out-of-plane plate modes can hardly be overlapped with each other in same frequency range due to the weak coupling between the resonator mode and the plate mode, and thus the broad complete bandgaps are difficult to obtain too. At the same time, a common feature of phononic-crystal plates is matrix structures made of nonmetallic materials. However, most mechanical structures typically consist of metals, and a phononiccrystal plate with a nonmetallic material matrix can hardly be used for vibration and noise reduction in mechanical engineering. Recently, a metal-matrix embedded phononic crystal which consists of periodic double-sided stepped resonators deposited on a two-dimensional phononic plate within a steel matrix was proposed. It is found that the bandwidth is increased, but the opening location of the bandgap is higher (1000 Hz) [19]. In other words, the extension of the bandwidth into the lower frequency range remains a challenging task for a metal-matrix

The existence of low-frequency locally resonant bandgaps (LRBGs) gives rise to the application of locally resonant phononic crystals (LRPCs) in the reduction of low-frequency vibration and noise [3]. However, it has been a challenging task due to the long wavelength and weak attenuation of the low-frequency waves. In the past years, many LRPCs were proposed to obtain bandgaps in low frequency range or broaden bandgaps in the low frequency range [4–19]. More recently, the propagation of Lamb waves (elastic waves) in phononic crystal plates, which is based on a local resonance mechanism, has attracted attention due to their potential applications in filters, resonators, waveguides, and for vibration insulation. In general, phononiccrystal plates can be classified into two types according to their structural features: flat plates and stubbed plates. The flat phononic-crystal plate consists of a periodical array of holes or periodic inclusions of foreign material in a homogeneous plate [5]. Many previous studies focused on this type of phononic-crystal plates to investigate the bandgap properties. It is not easy to obtain lower frequency phononic bandgaps using a flat phononic-crystal plate. Stubbed phononic-crystal plates can be classified into two types: the single-sided stubbed phononic-crystal plate and the doublesided stubbed phononic-crystal plate. The single-sided stubbed phononic-crystal plate [3] consists of a square array of stubs on one side of a homogeneous plate [6]. Subsequently, many researchers investigated the effects of material properties and geometric parameters on the BG of the single-sided stubbed phononic-crystal plate and found that the mass and the geometric parameter effect can help tune the bandgap into low frequencies [8]. Later, a novel single-sided stubbed phononiccrystal plate was proposed. It consists of a square array of stubs on one side of a two-dimensional binary locally resonant phononic plate. It can increase of the relative bandwidth over the classical single-sided stubbed plate [12]. The double-sided stubbed phononic-crystal plate consists of a square array of stubs on both sides of a homogeneous plate. It can reduce the bandgap compared to a single-sided stubbed phononic-crystal plate [9]. Recently, a novel double-sided stubbed phononic-crystal plate was proposed. Unlike the classical double-sided stubbed phononic-crystal

**30**

embedded phononic crystal.

Thus, how to adjust the in-plane and out-of-plane gaps overlapping with each other in the lower frequencies (below 100 Hz) or increasing the in-plane and outof-plane gaps in the same range is an important issue for the control of the vibration in a practical case.

In this chapter, two kinds of metal-matrix embedded phononic crystals are introduced. The one is called the lower frequency complete bandgap metal-matrix embedded phononic crystals [14] because it can produce lower frequency complete bandgap, and the other is called the broad complete bandgap metal-matrix embedded phononic crystals [20] because it can produce broad complete bandgap.

## **2. The method of calculation**

In order to investigate the band properties of the metal-matrix embedded phononic crystals, a series of calculations on dispersion relations and transmission spectra are conducted with FEM based on the Bloch theorem. For the calculation of the dispersion relations, the governing field equations for elastic wave propagation in solids are given by

$$\sum\_{j=1}^{2} \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \sum\_{l=1}^{2} \sum\_{k=1}^{3} c\_{ijkl} \frac{\partial u\_{k}}{\partial \mathbf{x}\_{l}} \right) = \rho \frac{\partial^{2} u\_{l}}{\partial t^{2}} \text{ ( $i = 1, 2, 3$ )}\tag{1}$$

where ρ is the mass density, t is the time, ui is the displacement, cijkl are the elastic constants, and xj (j = 1, 2, 3) represents the coordinate variables x, y, and z, respectively. According to the Bloch theorem, in the FEM formulation, the displacement field can be expressed as

$$u(r) = e^{i(kr)} u\_k(r) \tag{2}$$

where u is the displacement at the nodes and r is the position vector located at the boundary nodes. The Bloch wave vector k = (kx, ky) is the wave vector limited to the first Brillouin zone of the reciprocal lattice. Since the metal-matrix embedded phononic crystal plate is periodic in the xy-direction and finite in the z-direction, only the unit cell (shown as in **Figure 1(b)** or **Figure 3(b)**) needs to be considered in the FEM calculation when the Bloch theorem is adopted on the boundaries between the unit cell and its two adjacent cells, given by

$$u\_i(\mathcal{X} + a, \mathcal{y} + a) = e^{i\left(k\_i a \star k\_j a\right)} u\_i(\mathcal{X}, \mathcal{y}) \left(i = \mathcal{X}, \mathcal{y}, x\right) \tag{3}$$

#### **Figure 1.**

*(a and b) Schematic of the part and the unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals, respectively. (c) The corresponding first irreducible Brillouin zone (red region) and the high-symmetry M, Γ, and X.*

where the elastic displacement vector is denoted by *u*; the position vectors are denoted by *x*, *y*, and *z*; and *kx* and *ky* are the Bloch wave vectors limited in the irreducible first Brillouin zone (shown as in **Figure 1(c)** or **Figure 3(c)**). The Bloch calculation gives the eigenfrequencies and the corresponding eigenvectors, and then the dispersion relationships can be obtained by changing the wave vector in the first irreducible Brillouin zone.

To further demonstrate the existence of the bandgaps of the metal-matrix embedded phononic crystals, the transmission spectra for a structure with finite units along the x- or y-direction is calculated by using FEM. The acceleration excitation source is incident from one side (left side) of the finite structure and propagates along the *x*- or *y*-direction. The corresponding transmitted acceleration is recorded on another side (right side) of the structure. The transmission spectrum is defined as

$$TL = \mathbf{10} \log \left(\frac{\alpha\_o}{\alpha\_i}\right) \tag{4}$$

where *αo* and *αi* are the output and input accelerations of the metal-matrix embedded phononic crystals, respectively. Finally, the transmission spectra can be obtained by changing the excitation frequency of the incident acceleration.
