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

#### Figure 6.

Not only the thickness of the defect layer has an obvious effect on wave localization but also the type of the defect layer has a significant effect on the localized modes inside the band gaps. Although we introduced two different materials in Figure 6(c) and (d) with the same thickness, the number and width of the localized peaks had greatly changed. In Figure 6(c), we used Au which has mechanical constants higher than the host materials, while in Figure 5(d), we used nylon which has mechanical constants lower than the host materials. As a result, the width and number of the transmission bands are larger in Figure 6(d) than in Figure 6(c). Actually, the nylon is a very soft material like a spring and can introduce more

The dispersion curve of SH-waves propagating in the defected PnC. A defect layer from aluminum with thickness ad = aA was immersed between the two periodic unit cells (stop-bands shaded with the gray color).

Now we will investigate the effects of temperature elevation on the PnC structure and phononic band gaps. Two temperature degrees (T = 35 and 180°

were considered. Figure 7 shows the response of the dispersion relations with

C)

resonant modes inside the PnC structure.

Figure 4.

Figure 5.

20

A schematic diagram of a defect 1D binary PnC.

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3.1.3 Temperature effects on the band structure of PnC

The calculated dispersion curve of SH-waves propagating in a 1D binary defected PnC. (a) ad = 2aA (Al), (b) ad = 4aA (Al), (c) ad = aA (Au), and (d) ad = aA (Nylon) (stop-bands shaded with the gray color).

different temperatures at SH-waves propagation through PnCs. We can note in Figure 7(a) and (b) that, with increasing the temperature from T ¼ 35 to <sup>T</sup> <sup>¼</sup> <sup>180</sup>° C, respectively, the pass/stop band width remains constant due to the opposite increment in the thermal stress, which has a negative value and maintains materials dimensions constant. Therefore, there is not any variation in the thickness

#### Figure 7.

The dispersion curve of SH-waves propagates in a PnC from lead and epoxy materials. Different temperatures are considered. (a) <sup>T</sup> <sup>¼</sup> 35° <sup>C</sup> and (b) <sup>T</sup> <sup>¼</sup> <sup>180</sup>° C (stop-bands shaded in gray).

of any material, which in turn keeps the width of the band gaps constant. Finally, we can note only band edges changed slightly with increasing the temperatures. Such small shift in the band gaps is related to the different thermal expansion coefficients of the two materials. Based on this result, the effects of temperature on the PnC structure are greatly related to the type of waves that propagate through crystal structure. Hence, we will verify this result by studying the temperature effects on PnCs at plane wave propagation as well.

### 3.2 Plane waves results

As depicted in Figure 1, we can calculate the reflection coefficient of S- and Pwaves in the x-direction through the PnC structure. The PnC structure is proposed to be bonded between two semi-infinite materials (nylon material) at the two ends. The subscripts "0" and "e" denote the left and the right of the PnC structure, respectively.

The reflection coefficient of the displacement field through a PnC structure is given by the form [39],

$$\frac{U\_1}{U\_0} = \frac{T\_{12} + E\_0 T\_{11} - E\_0 E\_\epsilon T\_{21} - E\_\epsilon T\_{22}}{E\_0 (T\_{11} - E\_\epsilon T\_{21}) - (T\_{12} - E\_\epsilon T\_{22})},\tag{26}$$

The frequency regions at R ≈ 1 of P- and S-waves represent the complete band gaps of the plane waves through the PnC structure. There is only a fine difference in the band gaps edges between Figure 8 and those obtained by the dispersion relation. In Figure 8, the relation is plotted between the reflectance and <sup>ω</sup>a=2PicT , and cT <sup>¼</sup> cSB is chosen for both P- and S-waves reflectance. Therefore, we will use the reflection coefficient to describe the effects of the defect layer and temperature instead of dispersion relations in order to plot both P- and S-waves in the same graph and for

3.2.1 Influences of the defect layer/temperature on the phononic band gaps "plane wave"

In this section, we will study the effects of the defect layer and temperatures on the band structure of PnCs at the propagation of plane waves and compare with

First, we will use the same defected structure used in Section 3.1.2 with the same materials and conditions, only the angle of incidence will be maintained at <sup>θ</sup><sup>0</sup> <sup>¼</sup> 0°

C and 190°

C were considered in

Figure 9 confirms the last results of the effects of the defect layer on the localization modes through the PnC at the plane wave propagation. In Figure 9(b), a number of the localized waves was generated inside the phononic band gaps and was increased

order to illustrate the effects of temperature on the phonic band gaps. We noticed that the phononic band gaps were affected slightly by temperatures at plane wave propagation higher than SH-waves. Therefore, the propagation of elastic waves and localized modes can be affected by temperature elevation. From Figure 10(a), we can note that the reflectance of the P-wave (Red lines) is moved toward the higher frequencies (i.e., band gap at ωa=2π cT = 3.5). Such displacement in the band gap

C in Figure 10(b). These temperature effects on the band gaps can be explained by two reasons. First, the P-wave velocity is increased according to Eq. (24) because the temperature has a direct effect on the elastic constants. Consequently, temperature makes a

.

the practical and industrial purpose as well.

3.2.1.1 Defect layer influences on band structure

by increasing the defect layer thickness as well.

3.2.1.2 Temperature influences on band structure

edges is quite noticeable at T = 190°

Figure 9.

23

In this section, the two temperatures T = 50°

(a) Aluminum defect layer with ad = aA, (b) Aluminum defect layer with ad = 4aA.

those investigated for SH-waves.

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

where U<sup>1</sup> is the reflected amplitude and Tij ¼ T ið Þ ; j are the elements of the total transfer matrix T ¼ TnTn�<sup>1</sup>…Tm…T1.

Figure 8 shows the relation between the reflectance R versus ωa=2PicT (ω is considered the angular frequency in this relation and cT ¼ cSB) for P-wave (red dashed lines) and S-wave (black solid lines) [48]. From Figure 8, we can determine the range of frequencies for which the phononic band gaps can be occurred (reflectance of P- and S-waves is high R ≈ 1). Such high reflectance within the different frequency ranges represents the frequency band gaps of P- and S-waves inside the PnC structure.

From Figure 8, it can be seen that the phononic band gaps described by the reflection coefficient are agreed with those described by the dispersion relations.

#### Figure 8.

The reflectance R versus ωa=2π cT of P-wave (red lines) and S-waves (blue lines) propagated normally through a 1D PnC structure consist of four unit cells. Each unit cell consists of lead and epoxy materials.

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

of any material, which in turn keeps the width of the band gaps constant. Finally, we can note only band edges changed slightly with increasing the temperatures. Such small shift in the band gaps is related to the different thermal expansion coefficients of the two materials. Based on this result, the effects of temperature on the PnC structure are greatly related to the type of waves that propagate through crystal structure. Hence, we will verify this result by studying the temperature

As depicted in Figure 1, we can calculate the reflection coefficient of S- and Pwaves in the x-direction through the PnC structure. The PnC structure is proposed to be bonded between two semi-infinite materials (nylon material) at the two ends. The subscripts "0" and "e" denote the left and the right of the PnC structure,

The reflection coefficient of the displacement field through a PnC structure is

<sup>¼</sup> <sup>T</sup><sup>12</sup> <sup>þ</sup> <sup>E</sup>0T<sup>11</sup> � <sup>E</sup>0EeT<sup>21</sup> � EeT<sup>22</sup> E0ð Þ� T<sup>11</sup> � EeT<sup>21</sup> ð Þ T<sup>12</sup> � EeT<sup>22</sup>

where U<sup>1</sup> is the reflected amplitude and Tij ¼ T ið Þ ; j are the elements of the total

Figure 8 shows the relation between the reflectance R versus ωa=2PicT (ω is considered the angular frequency in this relation and cT ¼ cSB) for P-wave (red dashed lines) and S-wave (black solid lines) [48]. From Figure 8, we can determine the range of frequencies for which the phononic band gaps can be occurred (reflectance of P- and S-waves is high R ≈ 1). Such high reflectance within the different frequency ranges represents the frequency band gaps of P- and S-waves inside the

From Figure 8, it can be seen that the phononic band gaps described by the reflection coefficient are agreed with those described by the dispersion relations.

The reflectance R versus ωa=2π cT of P-wave (red lines) and S-waves (blue lines) propagated normally through

a 1D PnC structure consist of four unit cells. Each unit cell consists of lead and epoxy materials.

, (26)

effects on PnCs at plane wave propagation as well.

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U<sup>1</sup> U<sup>0</sup>

transfer matrix T ¼ TnTn�<sup>1</sup>…Tm…T1.

3.2 Plane waves results

given by the form [39],

respectively.

PnC structure.

Figure 8.

22

The frequency regions at R ≈ 1 of P- and S-waves represent the complete band gaps of the plane waves through the PnC structure. There is only a fine difference in the band gaps edges between Figure 8 and those obtained by the dispersion relation. In Figure 8, the relation is plotted between the reflectance and <sup>ω</sup>a=2PicT , and cT <sup>¼</sup> cSB is chosen for both P- and S-waves reflectance. Therefore, we will use the reflection coefficient to describe the effects of the defect layer and temperature instead of dispersion relations in order to plot both P- and S-waves in the same graph and for the practical and industrial purpose as well.
