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

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 example, the stop bands for θ<sup>0</sup> = 15° become a pass band for θ<sup>0</sup> = 45° . With increasing 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 appeared when the angle θ<sup>0</sup> reached the value 85° . The stop band increased to be the entire band structure of the PnC, and no propagation bands were observed. The explanation of such phenomenon is deduced from the parameter 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 ; if <sup>c</sup> cj . become smaller than sin θ0,the displacement field 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 to propagate through the PnC structure.

## 3.1.2 Defective mode effect on the band structure of the PnC

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 within the defect layer.

#### Figure 3.

The calculated dispersion curves of SH-waves in a 1D binary perfect PnC at the incident angles (a) θ<sup>0</sup> ° = 30° , (b) θ<sup>0</sup> ° = 50°, and (c) θ<sup>0</sup> ° = 70° .

#### Figure 4.

A schematic diagram of a defect 1D binary PnC.

#### Figure 5.

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).

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

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).

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

The dispersion curve of SH-waves propagates in a PnC from lead and epoxy materials. Different temperatures

C (stop-bands shaded in gray).

<sup>C</sup> and (b) <sup>T</sup> <sup>¼</sup> <sup>180</sup>°

<sup>T</sup> <sup>¼</sup> <sup>180</sup>°

Figure 7.

21

are considered. (a) <sup>T</sup> <sup>¼</sup> 35°

Figure 6.

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

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 resonant modes inside the PnC structure.

#### 3.1.3 Temperature effects on the band structure of PnC

Now we will investigate the effects of temperature elevation on the PnC structure and phononic band gaps. Two temperature degrees (T = 35 and 180° C) were considered. Figure 7 shows the response of the dispersion relations with
