Abstract

In this work, we demonstrate a comprehensive theoretical study of onedimensional perfect and defect phononic crystals. In our study, we investigate the elastic and shear waves with the influences of thermal effects. The numerical calculations based on the transfer matrix method (TMM) and Bloch theory are presented, where the TMM is obtained by applying the continuity conditions between two consecutive sub-cells. Also, we show that by introducing a defect layer in the perfect periodic structures (defect phononic crystals), we obtain localization modes within the band structure. These localized modes can be implemented in many applications such as impedance matching, collimation, and focusing in acoustic imaging applications. Then, we investigate the influences of the incident angle and material types on the number and intensity of the localized modes in both cases of perfect/defect crystals. In addition, we have observed that the temperature has a great effect on the wave localization phenomena in phononic band gap structures. Such effects can change the thermal properties of the PnCs structure such as thermal conductivity, and it can also control the thermal emission, which is contributed by phonons in many engineering structures.

Keywords: thermal emission, dispersion relation, phononic band gap, localized modes

### 1. Introduction

Phononic crystals (PnCs) are new composite materials which can interact, manipulate, trap, prohibit, and transmit the propagation of mechanical waves. Recently, great efforts have been dedicated to study these novel materials to be used in many potential engineering applications. By reference to the meaning of idiomatic, the term "phononic" was derived in analogy to the term "phonon," which is considered as a quantization of the lattice vibrations. In the previous conscious of science, these vibrations are impossible to be controlled because the atoms in the solid cannot move independently of each other because they are connected by chemical bonds. By appearing PnCs, this assumption was changed and the mechanical wave can be filtered, transmitted, stopped, and localized within specific frequencies called phononic band gaps [1, 2].

Within the phononic band gaps, the mechanical waves of all types are greatly blocked. Actually, the formation of the phononic band gaps is back to the variation in the mechanical properties of the materials that build the PnC structure. Therefore, Bragg interference at the interface between each two materials can be

obtained. As a result, the phononic band gaps and transmission bands are formed. Moreover, novel properties such as negative refraction and acoustic metamaterials are presented in PnC structures [3–7]. These novel properties of PnCs can be utilized in many industrial and engineering applications such as MEMS applications, filters, waveguides, clocking, multiplexers, and sensor applications [8–12].

mechanism has been implemented similarly in photonic crystals to slow light. Consequently, the produced waveguides in PnCs structures can be used in focusing and collimation of acoustic waves in medical ultrasound applications, sensors, and

In the present work, we introduce the formation of phononic band gaps under the influences of many physical parameters. First, the general case of SH-wave propagation in arbitrary direction will be investigated by using TMM with calculation of the dispersion curves as well. Also, the influences of the incident angle on the band gaps are analyzed and discussed. In addition, this work focuses on correlating and comparing the results of SH-waves with in-plane waves propagating normally to the structure. For the in-plane waves, the reflection coefficients for S- and Pwaves are plotted and compared with the dispersion relations curves. Furthermore, we are demonstrating the wave localization phenomenon in PnCs and the effects of the temperature on the band structure of PnCs and the localized modes for both in-plane and SH-waves. Also, the numerical results are presented and discussed to investigate the effect of the defect layer on the wave localization modes inside the structures. Finally, the effects of the thickness and type of the defect layer material

Figure 1 shows the schematic diagram of the 1D PnC crystal structure. The proposed crystal structure has an infinite number of the periodically arranged unit cells. The unit cell may include two or more layers; here, we propose that it is made by only two materials A and B, respectively. The two materials are labeled by the subscript j ¼ 1, 2. Also, the thickness of the unit cell (the lattice constant) is a ¼ a<sup>1</sup> þ a2: The thickness, Lame' constant, shearing modulus, Poisson's ratios, mass density, and Young's modulus of the two layers are denoted by equation

<sup>1</sup> � <sup>2</sup>ν<sup>j</sup>

 =ν<sup>j</sup> , respectively.

<sup>∂</sup>x<sup>2</sup> <sup>¼</sup> <sup>ρ</sup><sup>j</sup>

is the displacement components along the <sup>z</sup>-direction, <sup>t</sup> is the time,

where T is the temperature variation, β<sup>j</sup> is the thermal expansion coefficients,

The governing equation of anti-plane shear waves (SH-waves) polarized in the z-direction propagated in the xy-plane can be written in the following form [45–47].

φ€<sup>j</sup> xj; yj

; t

ð Þ <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup> , (1)

MEMS applications [2, 41–44].

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

on the band gap structure had been discussed.

2.1 Equation of wave propagation

, νj, ρj, Ej Ej ¼ λ<sup>j</sup> 1 þ ν<sup>j</sup>

; t <sup>þ</sup> <sup>σ</sup>tx

A schematic diagram of a perfect 1D binary PnC structure.

j ∂2 φ2 <sup>j</sup> xj; yj ; t 

aj, λj, μ<sup>j</sup>

φ<sup>j</sup> xj; yj

Figure 1.

13

μj ∇2

; t

φ<sup>j</sup> xj; yj

2. Theoretical analysis and numerical models

As it is well-known, the physical origin of all crystal structures has the same idea of the design. Photonic crystals and semiconductors devices could not develop such types of the previous applications. From the scientific point of view, we should mention that the physical nature of PnCs is different from one of photonic crystals, as well as semiconductors. Generally, the various forms of waves are referred to as: electron waves as scalar waves and optical waves as vector waves, while elastic waves as tensor waves [13–19].

As a result, to design PnCs having a complete phononic band gap, the mechanical properties must be changed not only in one direction (i.e. x- direction) but also in all three directions of space (i.e., x, y, and z directions). Therefore, PnCs can be classified according to the periodicity into three types, i.e., the one-dimensional (1D), the two-dimensional (2D), and the three-dimensional (3D) PnCs [20–24]. PnCs can control the entire spectrum of phonons frequency from 1 Hz to THz range. As it is well-known, the waves that propagate through solid media are called elastic waves, while those that propagate through fluids are called acoustic waves. Also, PnCs can control the different types of mechanical waves such as elastic waves, acoustic waves, and surface waves. Therefore, unlike photonic crystals, not all 3D PnCs possess complete phononic band gaps. PnCs must possess band gaps for both elastic and acoustic waves at the same frequency region. Consequently, we have to fabricate PnC structures for both solid and fluid media [25–28]. Hence after, these different polarizations introduce more challenges to PnCs higher than other crystal, which in turn makes the theoretical manipulation of PnCs more attractive and perspective.

From the previous observations of PnCs structures, several methods were developed to calculate the phononic band gaps such as Plane-Wave-expansion Method (PWM), Bloch-Floquet Method (BFM), and finite different time-domain method (FDTD) [29–33]. In this chapter, we will depend on the Transfer Matrix Method (TMM) [34] for calculating the reflection coefficient and the transmission coefficient of the one-dimensional PnC structure. Such method considered very suitably for the 1D structure due to its recursive nature, since it allows the continuity of the waveform at the interface between each two layers [35, 36]. Also, by using the TMM, we can obtain the dispersion relations of the mechanical waves through the periodic PnC structures. PnCs and temperature are in mutual influences in several ways [37]. For example, at any temperature, mainly the huge contribution in the thermal conductivity of many materials is dominant by phonon contribution, which is a function of phonon mean free path and the Boltzmann distribution of phonons of any material. Also, the thermal conductivity is depending on the thickness of a material, where optical branches contribute with about 30% of the thermal conductivity of any material. By inhibiting the acoustic phonon population, the optical phonon relaxation is indirectly inhibited by up to 30% and hence limits their contribution to thermal conductivity. Furthermore, in silicon PnC, the thermal contribution of phonons has been reduced to less than 4% of the value for bulk silicon at room temperature [38–40].

Moreover, the defected PnCs structures have wonderful application in wave guiding and multiplexing. But the produced structures are different from the proposed ideal structures due to the errors and defaults in manufacturing. By removing some layers or materials from the ordered periodic structure in a PnC, we can create a point or a waveguide defects that are able to localize and bend signals. Such

### Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

obtained. As a result, the phononic band gaps and transmission bands are formed. Moreover, novel properties such as negative refraction and acoustic metamaterials are presented in PnC structures [3–7]. These novel properties of PnCs can be utilized in many industrial and engineering applications such as MEMS applications, filters, waveguides, clocking, multiplexers, and sensor applications [8–12]. As it is well-known, the physical origin of all crystal structures has the same idea of the design. Photonic crystals and semiconductors devices could not develop such types of the previous applications. From the scientific point of view, we should mention that the physical nature of PnCs is different from one of photonic crystals, as well as semiconductors. Generally, the various forms of waves are referred to as: electron waves as scalar waves and optical waves as vector waves, while elastic

Photonic Crystals - A Glimpse of the Current Research Trends

As a result, to design PnCs having a complete phononic band gap, the mechanical properties must be changed not only in one direction (i.e. x- direction) but also in all three directions of space (i.e., x, y, and z directions). Therefore, PnCs can be classified according to the periodicity into three types, i.e., the one-dimensional (1D), the two-dimensional (2D), and the three-dimensional (3D) PnCs [20–24]. PnCs can control the entire spectrum of phonons frequency from 1 Hz to THz range. As it is well-known, the waves that propagate through solid media are called elastic waves, while those that propagate through fluids are called acoustic waves. Also, PnCs can control the different types of mechanical waves such as elastic waves, acoustic waves, and surface waves. Therefore, unlike photonic crystals, not all 3D PnCs possess complete phononic band gaps. PnCs must possess band gaps for both elastic and acoustic waves at the same frequency region. Consequently, we have to fabricate PnC structures for both solid and fluid media [25–28]. Hence after, these different polarizations introduce more challenges to PnCs higher than other crystal, which in turn makes the theoretical manipulation of PnCs more attractive

From the previous observations of PnCs structures, several methods were developed to calculate the phononic band gaps such as Plane-Wave-expansion Method (PWM), Bloch-Floquet Method (BFM), and finite different time-domain method (FDTD) [29–33]. In this chapter, we will depend on the Transfer Matrix Method (TMM) [34] for calculating the reflection coefficient and the transmission coefficient of the one-dimensional PnC structure. Such method considered very suitably for the 1D structure due to its recursive nature, since it allows the continuity of the waveform at the interface between each two layers [35, 36]. Also, by using the TMM, we can obtain the dispersion relations of the mechanical waves through the periodic PnC structures. PnCs and temperature are in mutual influences in several ways [37]. For example, at any temperature, mainly the huge contribution in the thermal conductivity of many materials is dominant by phonon contribution, which is a function of phonon mean free path and the Boltzmann distribution of phonons of any material. Also, the thermal conductivity is depending on the thickness of a material, where optical branches contribute with about 30% of the thermal conductivity of any material. By inhibiting the acoustic phonon population, the optical phonon relaxation is indirectly inhibited by up to 30% and hence limits their contribution to thermal conductivity. Furthermore, in silicon PnC, the thermal contribution of phonons has been reduced to less than 4% of the value for bulk

Moreover, the defected PnCs structures have wonderful application in wave guiding and multiplexing. But the produced structures are different from the proposed ideal structures due to the errors and defaults in manufacturing. By removing some layers or materials from the ordered periodic structure in a PnC, we can create a point or a waveguide defects that are able to localize and bend signals. Such

waves as tensor waves [13–19].

and perspective.

silicon at room temperature [38–40].

12

mechanism has been implemented similarly in photonic crystals to slow light. Consequently, the produced waveguides in PnCs structures can be used in focusing and collimation of acoustic waves in medical ultrasound applications, sensors, and MEMS applications [2, 41–44].

In the present work, we introduce the formation of phononic band gaps under the influences of many physical parameters. First, the general case of SH-wave propagation in arbitrary direction will be investigated by using TMM with calculation of the dispersion curves as well. Also, the influences of the incident angle on the band gaps are analyzed and discussed. In addition, this work focuses on correlating and comparing the results of SH-waves with in-plane waves propagating normally to the structure. For the in-plane waves, the reflection coefficients for S- and Pwaves are plotted and compared with the dispersion relations curves. Furthermore, we are demonstrating the wave localization phenomenon in PnCs and the effects of the temperature on the band structure of PnCs and the localized modes for both in-plane and SH-waves. Also, the numerical results are presented and discussed to investigate the effect of the defect layer on the wave localization modes inside the structures. Finally, the effects of the thickness and type of the defect layer material on the band gap structure had been discussed.

## 2. Theoretical analysis and numerical models

#### 2.1 Equation of wave propagation

Figure 1 shows the schematic diagram of the 1D PnC crystal structure. The proposed crystal structure has an infinite number of the periodically arranged unit cells. The unit cell may include two or more layers; here, we propose that it is made by only two materials A and B, respectively. The two materials are labeled by the subscript j ¼ 1, 2. Also, the thickness of the unit cell (the lattice constant) is a ¼ a<sup>1</sup> þ a2: The thickness, Lame' constant, shearing modulus, Poisson's ratios, mass density, and Young's modulus of the two layers are denoted by equation aj, λj, μj, νj, ρ<sup>j</sup> , Ej Ej ¼ λ<sup>j</sup> 1 þ ν<sup>j</sup> <sup>1</sup> � <sup>2</sup>ν<sup>j</sup> =ν<sup>j</sup> , respectively.

The governing equation of anti-plane shear waves (SH-waves) polarized in the z-direction propagated in the xy-plane can be written in the following form [45–47].

$$
\mu\_j \nabla^2 \rho\_j \left( \mathbf{x}\_j, \mathbf{y}\_j, t \right) + \sigma\_j^{\text{tr}} \frac{\partial^2 \rho\_j^2 \left( \mathbf{x}\_j, \mathbf{y}\_j, t \right)}{\partial \mathbf{x}^2} = \rho\_j \ddot{\rho}\_j \left( \mathbf{x}\_j, \mathbf{y}\_j, t \right) \quad (j = 1, 2), \tag{1}
$$

where T is the temperature variation, β<sup>j</sup> is the thermal expansion coefficients, φ<sup>j</sup> xj; yj ; t is the displacement components along the <sup>z</sup>-direction, <sup>t</sup> is the time,

Figure 1.

A schematic diagram of a perfect 1D binary PnC structure.

σtx <sup>j</sup> ¼ �Ej β<sup>j</sup> T= 1 � 2υ<sup>j</sup> � � is the thermal stress, and <sup>∇</sup><sup>2</sup> <sup>¼</sup> <sup>∂</sup>=∂x<sup>2</sup> <sup>j</sup> <sup>þ</sup> <sup>∂</sup>=∂y<sup>2</sup> <sup>j</sup> is the Laplacian operator. The solution φ<sup>j</sup> xj; yj ; t � �in the <sup>j</sup>th layer with time harmonic dependence can be expressed as [48],

$$\rho\_j(\mathbf{x}\_j, \mathbf{y}\_j, t) = \phi\_j(\mathbf{x}\_j) \exp\left[iky\_j \sin\theta\_0 - i\alpha t\right],\tag{2}$$

ϕð Þ<sup>i</sup> jL <sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup>

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

> τ ð Þi xzjL ¼ μ

be obtained as follows:

ð Þi jR <sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup>

exp iαqj

iαqj μj

where ν

forms,

T0 j ð Þ¼ 1; 1

T0 j ð Þ¼ 2; 1

15

<sup>j</sup> ð Þ <sup>0</sup> , <sup>ϕ</sup>ð Þ<sup>i</sup>

∂ϕð Þ<sup>i</sup> j a1∂ξ<sup>j</sup>

> ν ð Þi jR <sup>¼</sup> <sup>T</sup><sup>0</sup> j ν ð Þi

jR ; a<sup>1</sup> τ ð Þi xzjRo<sup>T</sup>

matrix of each unit cell. The four elements of T<sup>0</sup>

� � <sup>þ</sup> exp �iαqj

ζj

ν ð Þi <sup>2</sup><sup>R</sup> ¼ Ti ν

ν ð Þi <sup>1</sup><sup>L</sup> ¼ ν

ν ð Þi <sup>2</sup><sup>R</sup> ¼ Ti ν

� � � exp �iαqj

h i � �

ζj

j exp iαqj

2.3 Characteristic of the dispersion relation

obtained from Eq. (10) as follows:

written in the following form:

unit cell, the following condition is satisfied:

ð Þi j

jR <sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup>

and ν ð Þi jL <sup>¼</sup> <sup>ϕ</sup>ð Þ<sup>i</sup>

state wave vectors at right and left sides of each unit cell and T<sup>0</sup>

xzjLo<sup>T</sup> � �

ζj � � <sup>2</sup> , T<sup>0</sup>

j ð Þ¼ 1; 2

ζj

<sup>2</sup> , T<sup>0</sup>

At the interface between the layers, the following condition is satisfied:

Thus, the relationship between the right and left sides of the ith unit cell can be

ν ð Þi <sup>1</sup><sup>R</sup> ¼ ν ð Þi

ð Þi

where Ti is the accumulative transfer matrix of the ith unit cell and can be

Ti <sup>¼</sup> <sup>T</sup><sup>0</sup>

ð Þ i�1

vectors of the ð Þ i � 1 th unit cells and the ith unit cell in the following form:

ð Þ i�1

2T0

At the interface between the right side of the unit cell and the left side of the ith

By equating Eqs. (13) and (15), we can obtain the relationship between the state

ð Þ 0 , τ ð Þi xzjR ¼ μ

<sup>j</sup> ζ<sup>j</sup> � �,

> ð Þi j

where the subscripts L and R denote the left and right sides of the two layers, and 0≤ξ<sup>j</sup> ≤ ζ<sup>j</sup> ¼ aj=a<sup>1</sup> ð Þ j ¼ 1; 2 are the dimensionless thicknesses of materials A and B. Substituting Eqs. (7) and (8) into Eq. (9), the following matrix equation can

∂ϕð Þ<sup>i</sup> j a1∂ξ<sup>j</sup>

ζj

jL ; a<sup>1</sup> τ ð Þi

� � ð Þ <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; :…; <sup>n</sup> ,

jL ð Þ j ¼ 1; 2; i ¼ 1; 2; …; n , (10)

exp iαqj

j

are the dimensionless

<sup>j</sup> can be written in the following

ζj

ð Þ¼ <sup>2</sup>; <sup>2</sup> <sup>T</sup><sup>0</sup>

<sup>2</sup><sup>L</sup>: (12)

<sup>1</sup>: (14)

<sup>1</sup><sup>L</sup> ð Þ i ¼ 1; 2; …; n , (13)

<sup>2</sup><sup>R</sup> ð Þ i ¼ 2; ::……; n : (15)

<sup>2</sup><sup>R</sup> ð Þ i ¼ 2; ::…; n , (16)

� � � exp �iαqj

2iαqj μj

> j ð Þ 1; 1 :

<sup>j</sup> is the 2 � 2 transfer

ζj � �

(11)

<sup>j</sup> ,

(9)

Where, c is the velocity component of the incident wave, i <sup>2</sup> ¼ �1, <sup>k</sup> <sup>¼</sup> <sup>ω</sup>=<sup>c</sup> is the wave number, θ<sup>0</sup> is the angle of the incident wave, ω is the angular frequency, and ϕ<sup>j</sup> is the amplitude.

In many cases, it is more convenient to represent the thickness value by the following dimensionless coordinates:

$$\xi\_{\dot{j}} = \mathbb{X}\_{\dot{\mid a\_1 \bullet}}^{\omega\_{\dot{\mid a\_1}}} \eta\_{\dot{j}} = \mathbb{Y}\_{\dot{\mid a\_1}}^{\omega\_{\dot{\mid a\_1}}} (\boldsymbol{j} = \mathbf{1}, \mathbf{2}), \tag{3}$$

where a<sup>1</sup> is the average thickness value of the material A. a<sup>1</sup> is equal to a<sup>1</sup> for the periodic structures. By inserting Eq. (3) in Eqs. (1) and (2), we can obtain the following wave equations:

$$\rho\_j(\xi\_j, \eta\_j, t) = \phi\_j(\xi\_j) \exp\left[i\alpha \eta\_j \sin \theta\_0 - i\alpha t\right],\tag{4}$$

$$\left(1+\chi\_j\right)\frac{d^2\phi\_j}{d\xi\_j^2} - a^2 \left(\sin^2\theta\_0 - \frac{a\_j^2}{a^2}\right)\phi\_j = 0,\tag{5}$$

where cj represents the wave velocity in each material, <sup>χ</sup><sup>j</sup> <sup>¼</sup> <sup>σ</sup>tx <sup>j</sup> =μ<sup>j</sup> is the stress and shearing modulus ratio, α ¼ ka<sup>1</sup> represents the SH-waves dimensionless wave number, α<sup>j</sup> ¼ kja<sup>1</sup> is the dimensionless wave number and kj ¼ ω=cj is the wave vector of materials A and B, respectively. Eq. (5) has a general dimensionless solution, and it is given in the following form,

$$\phi\_j(\xi\_j) = A\_j \exp\left(-iaq\_j\xi\_j\right) + B\_j \exp\left(+iaq\_j\xi\_j\right), \quad (j=1,2), \tag{6}$$

where Aj and Bj are unknown coefficient to be determined, and qj is a parameter and has the value qj <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffi 1þχ<sup>j</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c<sup>2</sup>=c<sup>2</sup> <sup>j</sup> � sin <sup>2</sup>θ<sup>0</sup> q . Therefore, the dimensionless solution φ<sup>j</sup> ξj; η<sup>j</sup> ; t � � with time harmonic dependence is:

$$\rho\_j(\xi\_j, \eta\_j, t) = \left( A\_j \exp\left( -iaq\_j \xi\_j \right) + B\_j \exp\left( +iaq\_j \xi\_j \right) \right) \left( \exp\left[ ia\eta\_j \sin\theta\_0 - iat \right] \right). \tag{7}$$

#### 2.2 Transfer matrix method

The dimensionless shear stress component is given as follows:

$$
\pi\_{\text{xx}j} = \mu\_j \frac{\partial \rho\_j}{\overline{a}\_1 \partial \xi\_j} \quad (j = \mathbf{1}, \ \mathbf{2}). \tag{8}
$$

Assuming that the PnC consists of n unit cells, the boundary conditions at the left and right sides of the two layers in the ith unit cell can be written in the following form [45],

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

σtx

<sup>j</sup> ¼ �Ej β<sup>j</sup> T= 1 � 2υ<sup>j</sup>

ϕ<sup>j</sup> is the amplitude.

following wave equations:

Laplacian operator. The solution φ<sup>j</sup> xj; yj

φ<sup>j</sup> xj; yj

φ<sup>j</sup> ξj; η<sup>j</sup>

solution, and it is given in the following form,

¼ Aj exp �iαqj

ffiffiffiffiffiffiffi 1þχ<sup>j</sup>

¼ Aj exp �iαqj

q

with time harmonic dependence is:

ϕ<sup>j</sup> ξ<sup>j</sup> � �

and has the value qj <sup>¼</sup> <sup>1</sup>

; t � �

following form [45],

14

2.2 Transfer matrix method

φ<sup>j</sup> ξj; η<sup>j</sup>

; t � �

φ<sup>j</sup> ξj; η<sup>j</sup>

1 þ χ<sup>j</sup> � � <sup>d</sup><sup>2</sup>

; t � �

; t � �

Photonic Crystals - A Glimpse of the Current Research Trends

dependence can be expressed as [48],

following dimensionless coordinates:

� � is the thermal stress, and <sup>∇</sup><sup>2</sup> <sup>¼</sup> <sup>∂</sup>=∂x<sup>2</sup>

¼ ϕ<sup>j</sup> xj

Where, c is the velocity component of the incident wave, i

ξ<sup>j</sup> ¼ xj

; t � �

wave number, θ<sup>0</sup> is the angle of the incident wave, ω is the angular frequency, and

In many cases, it is more convenient to represent the thickness value by the

where a<sup>1</sup> is the average thickness value of the material A. a<sup>1</sup> is equal to a<sup>1</sup> for the

periodic structures. By inserting Eq. (3) in Eqs. (1) and (2), we can obtain the

� <sup>α</sup><sup>2</sup> sin <sup>2</sup>

and shearing modulus ratio, α ¼ ka<sup>1</sup> represents the SH-waves dimensionless wave number, α<sup>j</sup> ¼ kja<sup>1</sup> is the dimensionless wave number and kj ¼ ω=cj is the wave vector of materials A and B, respectively. Eq. (5) has a general dimensionless

¼ ϕ<sup>j</sup> ξ<sup>j</sup> � �

where cj represents the wave velocity in each material, <sup>χ</sup><sup>j</sup> <sup>¼</sup> <sup>σ</sup>tx

ξj � �

<sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c<sup>2</sup>=c<sup>2</sup>

ξj � �

� � � �

The dimensionless shear stress component is given as follows:

∂φj a1∂ξ<sup>j</sup>

left and right sides of the two layers in the ith unit cell can be written in the

Assuming that the PnC consists of n unit cells, the boundary conditions at the

τxzj ¼ μ<sup>j</sup>

ϕj dξ<sup>2</sup> j

� � exp ikyj sin <sup>θ</sup><sup>0</sup> � <sup>i</sup>ω<sup>t</sup>

h i

=a<sup>1</sup> , η<sup>j</sup> ¼ yj=a<sup>1</sup> ð Þ j ¼ 1; 2 , (3)

exp iαη<sup>j</sup> sin θ<sup>0</sup> � iωt h i

> <sup>θ</sup><sup>0</sup> � <sup>α</sup><sup>2</sup> j α2

!

þ Bj exp þiαqj

where Aj and Bj are unknown coefficient to be determined, and qj is a parameter

þ Bj exp þiαqj

ξj

<sup>j</sup> � sin <sup>2</sup>θ<sup>0</sup>

ξj � � <sup>j</sup> <sup>þ</sup> <sup>∂</sup>=∂y<sup>2</sup>

in the jth layer with time harmonic

<sup>j</sup> is the

, (2)

<sup>2</sup> ¼ �1, <sup>k</sup> <sup>¼</sup> <sup>ω</sup>=<sup>c</sup> is the

, (4)

<sup>j</sup> =μ<sup>j</sup> is the stress

ϕ<sup>j</sup> ¼ 0, (5)

, j ð Þ ¼ 1; 2 , (6)

. Therefore, the dimensionless solution

ð Þ j ¼ 1; 2 : (8)

exp iαη<sup>j</sup> sin θ<sup>0</sup> � iωt � � h i

: (7)

$$\begin{aligned} \phi\_{j\mathcal{L}}^{(i)} &= \phi\_{j}^{(i)}(\mathbf{0}), \ \phi\_{j\mathcal{R}}^{(i)} = \phi\_{j}^{(i)} \left( \boldsymbol{\zeta}\_{j} \right), \\ \boldsymbol{\pi}\_{\text{xzj}\mathcal{L}}^{(i)} &= \mu\_{j}^{(i)} \frac{\partial \phi\_{j}^{(i)}}{\overline{\mathbf{a}\_{1}} \partial \boldsymbol{\xi}\_{j}}(\mathbf{0}), \ \boldsymbol{\pi}\_{\text{xzj}\mathcal{R}}^{(i)} = \mu\_{j}^{(i)} \frac{\partial \phi\_{j}^{(i)}}{\overline{\mathbf{a}\_{1}} \partial \boldsymbol{\xi}\_{j}} \left( \boldsymbol{\zeta}\_{j} \right) \ (i = 1, 2, ..., n), \end{aligned} \tag{9}$$

where the subscripts L and R denote the left and right sides of the two layers, and 0≤ξ<sup>j</sup> ≤ ζ<sup>j</sup> ¼ aj=a<sup>1</sup> ð Þ j ¼ 1; 2 are the dimensionless thicknesses of materials A and B. Substituting Eqs. (7) and (8) into Eq. (9), the following matrix equation can be obtained as follows:

$$\nu\_{j\mathbb{R}}^{(i)} = T\_j' \nu\_{j\mathbb{L}}^{(i)} \quad (j = \mathbf{1}, 2; i = \mathbf{1}, 2, \dots, n), \tag{10}$$

$$\text{where } \nu\_{j\overline{\mathbb{R}}}^{(i)} = \left\{ \phi\_{j\overline{\mathbb{R}}}^{(i)}, \overline{\mathfrak{a}}\_{1} \boldsymbol{\tau}\_{\text{xx}\overline{\mathbb{R}}}^{(i)} \right\}^T \text{ and } \nu\_{j\overline{\mathbb{L}}}^{(i)} = \left\{ \phi\_{j\overline{\mathbb{L}}}^{(i)}, \overline{\mathfrak{a}}\_{1} \boldsymbol{\tau}\_{\text{xx}\overline{\mathbb{L}}}^{(i)} \right\}^T \text{ are the dimensionless.}$$

state wave vectors at right and left sides of each unit cell and T<sup>0</sup> <sup>j</sup> is the 2 � 2 transfer matrix of each unit cell. The four elements of T<sup>0</sup> <sup>j</sup> can be written in the following forms,

$$T\_j'(\mathbf{1}, \mathbf{1}) = \frac{\exp\left(iaq\_j\zeta\_j\right) + \exp\left(-iaq\_j\zeta\_j\right)}{2}, \ T\_j'(\mathbf{1}, \mathbf{2}) = \frac{\exp\left(iaq\_j\zeta\_j\right) - \exp\left(-iaq\_j\zeta\_j\right)}{2iaq\_j\omega\_j^j},$$

$$T\_j'(\mathbf{2}, \mathbf{1}) = \frac{iaq\_j\mu\_j'\left[\exp\left(iaq\_j\zeta\_j\right) - \exp\left(-iaq\_j\zeta\_j\right)\right]}{2}, \qquad T\_j'(\mathbf{2}, \mathbf{2}) = T\_j'(\mathbf{1}, \mathbf{1}). \tag{11}$$

#### 2.3 Characteristic of the dispersion relation

At the interface between the layers, the following condition is satisfied:

$$
\boldsymbol{\nu}\_{\mathbf{1}\mathbf{R}}^{(i)} = \boldsymbol{\nu}\_{\mathbf{2}\mathbf{L}}^{(i)}.\tag{12}
$$

Thus, the relationship between the right and left sides of the ith unit cell can be obtained from Eq. (10) as follows:

$$
\boldsymbol{\nu}\_{2\mathsf{R}}^{(i)} = T\_i \boldsymbol{\nu}\_{1\mathsf{L}}^{(i)} \quad (i = \mathtt{1}, 2, \ldots, n), \tag{13}
$$

where Ti is the accumulative transfer matrix of the ith unit cell and can be written in the following form:

$$T\_i = T\_2' T\_1'.\tag{14}$$

At the interface between the right side of the unit cell and the left side of the ith unit cell, the following condition is satisfied:

$$
\nu\_{1L}^{(i)} = \nu\_{2R}^{(i-1)} \quad (i = 2, \ldots, n). \tag{15}
$$

By equating Eqs. (13) and (15), we can obtain the relationship between the state vectors of the ð Þ i � 1 th unit cells and the ith unit cell in the following form:

$$
\boldsymbol{\nu}\_{2\mathbf{R}}^{(i)} = T\_i \boldsymbol{\nu}\_{2\mathbf{R}}^{(i-1)} \quad (i = 2, \ldots, n), \tag{16}
$$

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Therefore, Ti represents the transfer matrix between each two successive unit cells.

By using Floquet and Bloch theories, we can obtain the displacement and stress fields between each two neighboring unit cells (at the interface) by the following relations [46–49]:

$$
\phi\_{2R}^{(i)} = \phi\_{2R}^{(i-1)} \cdot \exp\left(ika\right), \quad \tau\_{\text{xx2R}}^{(i)} = \tau\_{\text{xx2R}}^{(i-1)} \cdot \exp\left(ika\right), \tag{17}
$$

where k is the wave number representing the direction of the traveled wave across the structure. Combining the above two equations leads to the following matrix equation:

$$
\boldsymbol{\nu}\_{2\mathbf{R}}^{(i)} = \ \boldsymbol{\nu}\_{2\mathbf{R}}^{(i-1)} \ \exp\left(i\boldsymbol{k}\boldsymbol{a}\right) \quad (i = 2, \ldots, n). \tag{18}
$$

By equating Eqs. (16) and (18) which leads to the following eigenvalue problem:

$$\left| T\_i - \left. \mathcal{e}^{ika} I \right| = 0. \tag{19}$$

From Eq. (18),

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

> v ð Þi <sup>2</sup><sup>R</sup> ¼ v

> v ð Þi <sup>2</sup><sup>R</sup> ¼ v

tion of strength magnitude proportional to the value kimaginary

<sup>λ</sup><sup>σ</sup> <sup>¼</sup> <sup>λ</sup><sup>θ</sup> <sup>þ</sup>

9β<sup>2</sup> B2 θ ρC<sup>ν</sup>

where the superscripts σ and θ indicate adiabatic and isothermal constants, β is the thermal expansion coefficient, B is the bulk modulusð Þ B ¼ λ þ 2=3μ , C<sup>ν</sup> is the specific heat at constant volume, θ is the absolute temperature, and ρ is the mass density. From Eq. 24, we can deduce that λ<sup>σ</sup> andλ<sup>θ</sup> are not the same and the differ-

In addition to the above relations, if the temperature is increased, not only the wave velocities will increase but also the thickness of each layer will change by the

where Δa is the thickness difference of any layer, ai is the original layer thickness, and Δt is the temperature difference. Hence after, these two variables will affect the longitudinal waves speed and the stress component in SH-waves equa-

> ffiffiffiffiffiffiffiffi λþ2μ ρ q

First, from a practical point of view, the number of layers in the 1D PnC structure should take a finite number. Therefore, we consider the unit cell of the PnC structure is made from two layers. The two layers are lead and epoxy materials

, μσ <sup>¼</sup> μθ

Δa ¼ β aiΔt, (25)

, which, in turn leads to the variation

2.5 Temperature influences on PnCs

ence between them should be considered.

tions. Since the P-wave velocity is cP ¼

of the band structure and band gaps properties.

3. Numerical examples and discussions

following relation [51–55],

following relation [53]:

3.1 SH-waves results

17

ð Þ i�1 <sup>2</sup><sup>R</sup> eika,

ð Þ i�1

case, the waves are prohibited from propagation in the PnC structure, which in turn, results in formation of the so-called phononic band gaps or stop bands.

<sup>2</sup><sup>R</sup> e�j j <sup>k</sup>imaginary <sup>a</sup>

In contrast to the above case, we can deduce from Eq. (23) that the displacement and stress at successive unit cells ð Þi th and ð Þ i � 1 th are the same and do not have a phase difference. Moreover, the incident wave has an exponential spatial attenua-

When a mechanical wave propagates through a PnC structure, resultant vibrations occur which may increase or decrease phonons motions. Therefore, it can heat or cool the PnC structure. Also, the thermal expansion of the constituents materials may change, which will affect the mechanical constants of the material as well. Since the vibrations occur very rapidly, there is no big chance to thermal energies to flow and the elastic constants measured by elastic waves propagation are changed adiabatically. The elastic constants are connected to the isothermal constants by the

:

� � � (23)

�. Hence after, at this

, (24)

Eq. (19) can be rewritten as follows:

$$T\_i \nu\_{2R}^{(i-1)} = \lambda \,\nu\_{2R}^{(i-1)},\tag{20}$$

where <sup>λ</sup> <sup>¼</sup> <sup>e</sup>ika is a complex eigenvalue and <sup>V</sup>ð Þ <sup>i</sup>�<sup>1</sup> <sup>2</sup><sup>R</sup> is a complex eigenvector.

#### 2.4 Band structure formation

The wave number in Eqs. (17) and (18) is a complex number, so it can be a positive or negative number; in general, the wave number can be written in the following form [50]:

$$k = k\_{\text{real}} - \text{ik}\_{\text{imaginary}} \tag{21}$$

where kreal and kimaginary are the real and imaginary wave numbers, respectively. Therefore, we can deduce from this relation that the complex wave number has two forms, so it can inhibit the incident waves within the phononic band gaps. Consequently, we have two frequency ranges and they are organized as follows:

$$\text{1. If } k = k\_{\text{real}} \text{ and } \; k\_{\text{real}} > 0.$$

From Eq. (18),

$$\begin{array}{lcl}\boldsymbol{\upsilon}\_{2\boldsymbol{R}}^{(i)} = & \boldsymbol{\upsilon}\_{2\boldsymbol{R}}^{(i-1)} \circ \mathbf{e}^{\mathrm{i}k\boldsymbol{a}},\\\boldsymbol{\upsilon}\_{2\boldsymbol{R}}^{(i)} = & \boldsymbol{\upsilon}\_{2\boldsymbol{R}}^{(i-1)} \circ \mathbf{e}^{\mathrm{i}|k\_{\mathrm{real}}|\boldsymbol{a}}.\end{array} \tag{22}$$

From Eq. (22), we can deduce that the displacement and stress at the successive unit cells ð Þ<sup>i</sup> th and ð Þ <sup>i</sup> � <sup>1</sup> th are differ only by a phase factor e<sup>i</sup> kreal j j<sup>a</sup> . At this condition, the elastic waves are allowed to propagate freely through the PnC structure with the corresponding frequencies and wave number, forming the so-called transmission bands.

$$\text{2. If } k = -\text{ik}\_{\text{imaginary}} \text{ and } \; k\_{\text{imaginary}} < 0.$$

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

From Eq. (18),

Therefore, Ti represents the transfer matrix between each two successive unit

By using Floquet and Bloch theories, we can obtain the displacement and stress fields between each two neighboring unit cells (at the interface) by the following

> ð Þi xz2<sup>R</sup> ¼ τ

where k is the wave number representing the direction of the traveled wave across the structure. Combining the above two equations leads to the following

By equating Eqs. (16) and (18) which leads to the following eigenvalue problem:

<sup>2</sup><sup>R</sup> <sup>¼</sup> λ νð Þ <sup>i</sup>�<sup>1</sup>

The wave number in Eqs. (17) and (18) is a complex number, so it can be a positive or negative number; in general, the wave number can be written in the

where kreal and kimaginary are the real and imaginary wave numbers, respectively. Therefore, we can deduce from this relation that the complex wave number has two forms, so it can inhibit the incident waves within the phononic band gaps. Conse-

> ð Þ i�1 <sup>2</sup><sup>R</sup> e<sup>i</sup>ka,

ð Þ i�1

tion, the elastic waves are allowed to propagate freely through the PnC structure with the corresponding frequencies and wave number, forming the so-called trans-

From Eq. (22), we can deduce that the displacement and stress at the successive

quently, we have two frequency ranges and they are organized as follows:

v ð Þi <sup>2</sup><sup>R</sup> ¼ v

v ð Þi <sup>2</sup><sup>R</sup> ¼ v

unit cells ð Þ<sup>i</sup> th and ð Þ <sup>i</sup> � <sup>1</sup> th are differ only by a phase factor e<sup>i</sup> kreal j j<sup>a</sup>

ð Þ i�1

<sup>2</sup><sup>R</sup> exp ð Þ ika ð Þ i ¼ 2; ::…; n : (18)

k ¼ kreal � ikimaginary, (21)

<sup>2</sup><sup>R</sup> <sup>e</sup><sup>i</sup> <sup>k</sup>real j j<sup>a</sup>: (22)

. At this condi-

¼ 0: (19)

<sup>2</sup><sup>R</sup> ; (20)

<sup>2</sup><sup>R</sup> is a complex eigenvector.

xz2<sup>R</sup> exp ð Þ ika , (17)

cells.

relations [46–49]:

matrix equation:

ϕð Þ<sup>i</sup>

<sup>2</sup><sup>R</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>i</sup>�<sup>1</sup>

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ν ð Þi <sup>2</sup><sup>R</sup> ¼ ν

Eq. (19) can be rewritten as follows:

2.4 Band structure formation

1. If k ¼ kreal and kreal > 0.

2. If k ¼ �ikimaginary and kimaginary <0

From Eq. (18),

mission bands.

16

following form [50]:

<sup>2</sup><sup>R</sup> exp ð Þ ika , τ

ð Þ i�1

Ti � e ikaI 

Ti ν ð Þ i�1

where <sup>λ</sup> <sup>¼</sup> <sup>e</sup>ika is a complex eigenvalue and <sup>V</sup>ð Þ <sup>i</sup>�<sup>1</sup>

$$\begin{array}{lcl}v\_{2R}^{(i)} = & v\_{2R}^{(i-1)} \text{ e^{ikat}},\\ v\_{2R}^{(i)} = & v\_{2R}^{(i-1)} \text{ e^{-}}|\_{\text{imayinary}}|\_{\text{.}}\end{array} \tag{23}$$

In contrast to the above case, we can deduce from Eq. (23) that the displacement and stress at successive unit cells ð Þi th and ð Þ i � 1 th are the same and do not have a phase difference. Moreover, the incident wave has an exponential spatial attenuation of strength magnitude proportional to the value kimaginary � � � �. Hence after, at this case, the waves are prohibited from propagation in the PnC structure, which in turn, results in formation of the so-called phononic band gaps or stop bands.

#### 2.5 Temperature influences on PnCs

When a mechanical wave propagates through a PnC structure, resultant vibrations occur which may increase or decrease phonons motions. Therefore, it can heat or cool the PnC structure. Also, the thermal expansion of the constituents materials may change, which will affect the mechanical constants of the material as well. Since the vibrations occur very rapidly, there is no big chance to thermal energies to flow and the elastic constants measured by elastic waves propagation are changed adiabatically. The elastic constants are connected to the isothermal constants by the following relation [51–55],

$$
\lambda^{\sigma} = \lambda^{\theta} + \frac{\Re \beta^2 B^2 \theta}{\rho \mathbf{C}\_{\nu}}, \quad \mu^{\sigma} = \mu^{\theta}, \tag{24}
$$

where the superscripts σ and θ indicate adiabatic and isothermal constants, β is the thermal expansion coefficient, B is the bulk modulusð Þ B ¼ λ þ 2=3μ , C<sup>ν</sup> is the specific heat at constant volume, θ is the absolute temperature, and ρ is the mass density. From Eq. 24, we can deduce that λ<sup>σ</sup> andλ<sup>θ</sup> are not the same and the difference between them should be considered.

In addition to the above relations, if the temperature is increased, not only the wave velocities will increase but also the thickness of each layer will change by the following relation [53]:

$$
\Delta \mathfrak{a} = \beta \mathfrak{a}\_i \Delta \mathfrak{t},\tag{25}
$$

where Δa is the thickness difference of any layer, ai is the original layer thickness, and Δt is the temperature difference. Hence after, these two variables will affect the longitudinal waves speed and the stress component in SH-waves equations. Since the P-wave velocity is cP ¼ ffiffiffiffiffiffiffiffi λþ2μ ρ q , which, in turn leads to the variation of the band structure and band gaps properties.

### 3. Numerical examples and discussions

#### 3.1 SH-waves results

First, from a practical point of view, the number of layers in the 1D PnC structure should take a finite number. Therefore, we consider the unit cell of the PnC structure is made from two layers. The two layers are lead and epoxy materials and denoted by the symbols A and B, respectively. Second, dispersion relations are plotted for an infinite number of unit cells because it depends on Bloch theory that manipulates the propagation of waves through infinite periodic structures. Therefore, the dispersive behavior of the periodic materials and structures with an arbitrary chosen unit cell configuration is considered as an example. The constants of the materials used in the calculations can be found in Refs. [46, 49] and in Table 1. Here, we consider the velocity c of the incident wave is 800 m/s and the angle of the incidence is <sup>θ</sup><sup>0</sup> <sup>¼</sup> <sup>20</sup>° . Also, the ambient temperature is proposed to be <sup>T</sup> <sup>¼</sup> <sup>20</sup>° C. The dispersive properties of the inhomogeneous structures are determined by the properties ratios of the constituent materials, here we consider the two materials thicknesses ratio as 1:1.

frequency regions that are plotted with the white color and represent the real valued wave number and pass bands. Second, the other frequency regions corresponding to

Figure 3(a) and (b) show the effects of the incident angle θ<sup>0</sup> on the band structures of the previous perfect structure. It can be seen that the wave propagation behavior of the perfect PnC changes obviously for different incident angles. For

value of the incident angle, the same stop bands became wider in the considered frequency regions. However, in Figure 3(c), we can note a wonderful phenomenon

the entire band structure of the PnC, and no propagation bands were observed.

will decay. Therefore, a total reflection to the elastic waves will be occurred, where all the incident wave energy is reflected back to the structure and no waves are allowed

As shown in Figure 4, the dispersion relation of the perfect PnC structure will differ than the defected ones. We consider a defect layer from aluminum (the defect layer thickness ad = aA i.e., 1/2 of any unit cell thickness) was immersed after the second unit cell, and the material properties are mentioned in Table 1. From

Figure 5, it was indicated that elastic waves can be trapped within the phononic band gap and the band structure changed significantly than the perfect ones. The width of the band gaps increased due to the increment of mismatch between the constituent materials, this back to the insertion of a second interface inside the PnC structure. Moreover, the defect layer acts as a trap inside the PnC structure, so a special wave frequency corresponding to that waveguide will propagate through the structure. Additionally, in Figure 6(a) (ad = 2aA) and Figure 6(b) (ad = 4aA), we studied the effects of the defect layer thickness on the position and number of the localized modes inside the band structure of the PnC. It was shown that the number and width of the localized modes were increased by increasing the defect layer thickness. Therefore, with increasing the thickness of the defect layer inside the periodic PnC structure, the localized modes within the band gap are strongly confined

The calculated dispersion curves of SH-waves in a 1D binary perfect PnC at the incident angles (a) θ<sup>0</sup>

. With increasing

° = 30° ,

. The stop band increased to be

become smaller than sin θ0,the displacement field

the imaginary wave number represent the phononic band gaps.

3.1.1 Influence of the angle of incidence on the phononic band gap

example, the stop bands for θ<sup>0</sup> = 15° become a pass band for θ<sup>0</sup> = 45°

The explanation of such phenomenon is deduced from the parameter

; if <sup>c</sup> cj .

3.1.2 Defective mode effect on the band structure of the PnC

appeared when the angle θ<sup>0</sup> reached the value 85°

<sup>j</sup> � sin <sup>2</sup>θ<sup>0</sup>

to propagate through the PnC structure.

<sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c<sup>2</sup>=c<sup>2</sup>

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

q

within the defect layer.

= 50°, and (c) θ<sup>0</sup>

° = 70° .

Figure 3.

(b) θ<sup>0</sup> °

19

qj <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffi 1þχ<sup>j</sup>

Figure 2 presents the dispersion relation of SH-waves in the first Brillouin zone, and it is plotted between the nondimensional frequency q ¼ ωa=cB and the nondimensional wave number ξ ¼ k � a, where cB is the wave velocity in the second material (epoxy). We considered the range of the nondimensional frequency is 0 ≤q≤ 9. From this figure, we can conclude that, first, there are some


#### Table 1. Material constants.

#### Figure 2.

The dispersion curve of SH-waves incident on the face of the 1D binary perfect PnC structure. Each unit cell consists of lead and epoxy materials (stop-bands shaded by the gray color).

### Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

and denoted by the symbols A and B, respectively. Second, dispersion relations are plotted for an infinite number of unit cells because it depends on Bloch theory that manipulates the propagation of waves through infinite periodic structures. Therefore, the dispersive behavior of the periodic materials and structures with an arbitrary chosen unit cell configuration is considered as an example. The constants of the materials used in the calculations can be found in Refs. [46, 49] and in Table 1. Here, we consider the velocity c of the incident wave is 800 m/s and the angle of the

The dispersive properties of the inhomogeneous structures are determined by the properties ratios of the constituent materials, here we consider the two materials

and it is plotted between the nondimensional frequency q ¼ ωa=cB and the nondimensional wave number ξ ¼ k � a, where cB is the wave velocity in the second material (epoxy). We considered the range of the nondimensional frequency is 0 ≤q≤ 9. From this figure, we can conclude that, first, there are some

> Shearing modulus <sup>μ</sup>�1010 (N/m<sup>2</sup> )

Figure 2 presents the dispersion relation of SH-waves in the first Brillouin zone,

Young's modulus <sup>E</sup> � <sup>10</sup><sup>10</sup> (N/m<sup>2</sup> )

Lead 11.4 3.3 0.54 1.536 0.43 29.5 0.128 Epoxy 1.180 0.443 0.159 0.435 0.368 22.5 1.182 Aluminum 2.699 6.1 2.5 6.752 0.355 23.9 0.9 Gold 19.32 15.0 2.85 8.114 0.42 14.2 0.13 Nylon 1.11 0.511 0.122 0.357 0.4 50 1.70

The dispersion curve of SH-waves incident on the face of the 1D binary perfect PnC structure. Each unit cell

consists of lead and epoxy materials (stop-bands shaded by the gray color).

Poisson's ratios ν

Thermal expansion coefficient <sup>β</sup> � <sup>10</sup>�<sup>6</sup> (1/° C)

. Also, the ambient temperature is proposed to be <sup>T</sup> <sup>¼</sup> <sup>20</sup>°

C.

Specific heat <sup>C</sup>ν�103 (J/kg. ° C)

incidence is <sup>θ</sup><sup>0</sup> <sup>¼</sup> <sup>20</sup>°

thicknesses ratio as 1:1.

Materials Mass

Table 1. Material constants.

Figure 2.

18

density <sup>ρ</sup>�10<sup>3</sup> (kg/ m<sup>3</sup> )

Lame' constant <sup>λ</sup>�10<sup>10</sup> (N/m<sup>2</sup> )

Photonic Crystals - A Glimpse of the Current Research Trends

frequency regions that are plotted with the white color and represent the real valued wave number and pass bands. Second, the other frequency regions corresponding to the imaginary wave number represent the phononic band gaps.
