**3.2 The broad complete BG phononic crystals**

### *3.2.1 The model of the phononic crystals*

The broad complete bandgap metal-matrix embedded phononic crystal [18] was composed of a square array of single "hard" cylinder stubs on both sides of a twodimensional binary locally resonant PC plate which composes of an array of rubber fillers embedded in the steel plate. **Figure 3(a)** and **(b)** shows part of the structure and its unit cell, respectively.

The metal-matrix embedded phononic crystals with single "hard" cylinder stubs which consist of "hard" stub such as steel stubs contact with rubber filler. The geometrical parameters of the structure are defined as follows: the diameter of the rubber filler, the steel plate thickness, and the lattice constant are denoted by *D*, *e*, and *a*, respectively; the height and the diameter of the stub are denoted by *h* and *d*, respectively. The material parameters used in the calculations are listed in **Table 1**.

#### *3.2.2 The results of the phononic crystals*

Through the use of the finite element method, the systems described in **Figure 3** were studied numerically. The band structures and displacement vector fields were computed according to the Bloch theorem. The single-unit cell (as shown in **Figure 3(b)**) is determined by the periodicity of the structure. The following structure parameters are used: *D* = 8 mm, *e* = 1 mm, *a* = 10 mm, *h* = 2.5 mm, and *d* = 7.5 mm.

There are 12 bands within 0–600 Hz (see **Figure 4(a)**). Besides the traditional plate modes, which are the in-plane modes (such as mode *S*2) and the out-of-plane modes (such as mode *A*2), many flat modes (such as modes *S*1, *A*1, *F*1), which are

#### **Figure 3.**

*(a and b) Schematic of the part and the unit cell of the broad 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.*

**35**

**Figure 4.**

*plane bandgaps, respectively.*

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

bands). The absolute bandwidth is 136 Hz.

the resonant modes of the single "hard" cylinder stubs, can be found. The bandgaps (one in-plane bandgap, one out-of-plane bandgap, and one complete bandgap) appear due to coupling between the two modes. The in-plane bandgap (**Figure 4(a)**: blue-dashed area) is due to coupling between the in-plane modes *S*2 and the corresponding flat mode *S*1. It ranges from 260 to 573 Hz (between the sixth and twelfth bands). The absolute bandwidth is 313 Hz. The out-of-plane bandgap (**Figure 4(a)**: green-dashed area) is due to coupling between the out-of-plane mode *A*2 and the corresponding flat modes *A*1. It ranges from 187 to 396 Hz (between the third and eighth bands), and the absolute bandwidth is 209 Hz; the complete bandgap (**Figure 4(a)**: red-dashed area) is due to the overlap between the in-plane bandgap and the outof-plane bandgap. It ranges from 260 to 396 Hz (between the sixth and eighth

*Band structures of (a) the broad complete bandgap metal-matrix embedded phononic crystals, (b) the transition PC plate, and (c) the classical PC plate. The insets are the schematic view of the unit cell of the corresponding structure. The red, blue, and yellow shadow regions denote the complete, in-plane, and out-of-*

As a comparison, we also calculated the band structures of the transition PC plate composed of double-sided composite cylinder stubs deposited on a twodimensional locally resonant phononic crystal plate that consists of an array of rubber fillers embedded in a steel plate and the classical PC plate which was proposed by Assouar [9]. They are shown in **Figure 4(b)** and **(c)**, respectively. For the classical phononic-crystal plate, its first complete bandgap (red-dashed area) is caused by the overlap between the second in-plane bandgap and the first out-of-plane bandgap. The associated absolute bandwidth is 29 Hz. For the transition phononic-crystal plate, the bandgaps are lowered after introducing the rubber filler, such that both the out-of-plane bandgap and in-plane bandgap were lowered. However, the out-of-plane bandgap is lowered more and overlaps with two in-plane bandgaps (the first and the second in-plane bandgaps). This causes the complete bandgaps to be increased, but the absolute bandwidth is also narrow (78 Hz). These phenomena confirm that the single "hard" cylinder stub has a special effect on the bandwidth and the bandwidth can be increased by introducing it. This occurs mainly because the in-plane bandgap is increased by introducing the single "hard"

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

**Figure 4.**

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

**3.2 The broad complete BG phononic crystals**

*3.2.1 The model of the phononic crystals*

*3.2.2 The results of the phononic crystals*

and its unit cell, respectively.

proposed structure gives rise to a significant lowering of the opening location of the first complete bandgap by a factor of 5.5 compared with the classical PC plate. Compared with the classical PC plate, introducing the double-sided composite taper stubs, the locations of both the in-plane and out-of-plane bandgaps are lowered, but the out-of-plane bandgap is always overlapped with the second in-plane bandgap; when introducing the rubber filler, the location of the in-plane bandgap is kept stationary, and the out-of-plane bandgap is shifted to lower frequency (59 Hz) overlapped with the first in-plane bandgap. Finally, a complete bandgap is generated in lower frequency (below 100 Hz). Therefore, the double-sided taper stub has a direct effect on the lowering of the location of the in-plane bandgaps (53 Hz), and the rubber filler has a direct effect on the lowering of the location of the out-ofplane bandgaps (59 Hz). It makes the out-of-plane bandgaps overlap with the first in-plane bandgap and leads to a complete bandgap in lower frequencies [14].

The broad complete bandgap metal-matrix embedded phononic crystal [18] was composed of a square array of single "hard" cylinder stubs on both sides of a twodimensional binary locally resonant PC plate which composes of an array of rubber fillers embedded in the steel plate. **Figure 3(a)** and **(b)** shows part of the structure

The metal-matrix embedded phononic crystals with single "hard" cylinder stubs which consist of "hard" stub such as steel stubs contact with rubber filler. The geometrical parameters of the structure are defined as follows: the diameter of the rubber filler, the steel plate thickness, and the lattice constant are denoted by *D*, *e*, and *a*, respectively; the height and the diameter of the stub are denoted by *h* and *d*, respectively. The material parameters used in the calculations are listed in **Table 1**.

Through the use of the finite element method, the systems described in **Figure 3** were studied numerically. The band structures and displacement vector fields were computed according to the Bloch theorem. The single-unit cell (as shown in **Figure 3(b)**) is determined by the periodicity of the structure. The following structure parameters

There are 12 bands within 0–600 Hz (see **Figure 4(a)**). Besides the traditional plate modes, which are the in-plane modes (such as mode *S*2) and the out-of-plane modes (such as mode *A*2), many flat modes (such as modes *S*1, *A*1, *F*1), which are

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

are used: *D* = 8 mm, *e* = 1 mm, *a* = 10 mm, *h* = 2.5 mm, and *d* = 7.5 mm.

**34**

**Figure 3.**

*high-symmetry M, Γ, and X.*

*Band structures of (a) the broad complete bandgap metal-matrix embedded phononic crystals, (b) the transition PC plate, and (c) the classical PC plate. The insets are the schematic view of the unit cell of the corresponding structure. The red, blue, and yellow shadow regions denote the complete, in-plane, and out-ofplane bandgaps, respectively.*

the resonant modes of the single "hard" cylinder stubs, can be found. The bandgaps (one in-plane bandgap, one out-of-plane bandgap, and one complete bandgap) appear due to coupling between the two modes. The in-plane bandgap (**Figure 4(a)**: blue-dashed area) is due to coupling between the in-plane modes *S*2 and the corresponding flat mode *S*1. It ranges from 260 to 573 Hz (between the sixth and twelfth bands). The absolute bandwidth is 313 Hz. The out-of-plane bandgap (**Figure 4(a)**: green-dashed area) is due to coupling between the out-of-plane mode *A*2 and the corresponding flat modes *A*1. It ranges from 187 to 396 Hz (between the third and eighth bands), and the absolute bandwidth is 209 Hz; the complete bandgap (**Figure 4(a)**: red-dashed area) is due to the overlap between the in-plane bandgap and the outof-plane bandgap. It ranges from 260 to 396 Hz (between the sixth and eighth bands). The absolute bandwidth is 136 Hz.

As a comparison, we also calculated the band structures of the transition PC plate composed of double-sided composite cylinder stubs deposited on a twodimensional locally resonant phononic crystal plate that consists of an array of rubber fillers embedded in a steel plate and the classical PC plate which was proposed by Assouar [9]. They are shown in **Figure 4(b)** and **(c)**, respectively. For the classical phononic-crystal plate, its first complete bandgap (red-dashed area) is caused by the overlap between the second in-plane bandgap and the first out-of-plane bandgap. The associated absolute bandwidth is 29 Hz. For the transition phononic-crystal plate, the bandgaps are lowered after introducing the rubber filler, such that both the out-of-plane bandgap and in-plane bandgap were lowered. However, the out-of-plane bandgap is lowered more and overlaps with two in-plane bandgaps (the first and the second in-plane bandgaps). This causes the complete bandgaps to be increased, but the absolute bandwidth is also narrow (78 Hz). These phenomena confirm that the single "hard" cylinder stub has a special effect on the bandwidth and the bandwidth can be increased by introducing it. This occurs mainly because the in-plane bandgap is increased by introducing the single "hard"

cylinder stub in the broad complete bandgap metal-matrix embedded phononic crystals. The absolute bandwidth of the in-plane bandgap is increased by a factor of 3.37 compared with a classical phononic-crystal plate [18].
