**3. The model and the results**

#### **3.1 The lower frequency complete BG phononic crystals**

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

The lower frequency bandgap metal-matrix embedded phononic crystal [14] was composed of a square array of composite taper stubs on both sides of a twodimensional binary locally resonant PC plate which composes an array of rubber fillers embedded in the steel plate. **Figure 1(a)** and **(b)** shows part of the proposed structure and its unit cell, respectively.

In the lower frequency bandgap metal-matrix embedded phononic crystals, the taper stub is composed of the *A* taper cap and the *B* which is located on the taper *A*. 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 taper stub are denoted by *h* (*hA* for taper *A* and *hB* for taper *B*) and *d* (the upper diameter of taper stub is denoted by *dup*, and the lower diameter is denoted by *dlow*), respectively. The material parameters used in the calculations are listed in **Table 1**. The taper *A* and the taper *B* are rubber and steel, respectively.

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

Through the use of the finite element method, the systems described in **Figure 1** were studied numerically. The band structures and displacement vector


**33**

**Figure 2.**

*plane bandgaps, respectively.*

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

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 = 5 mm

It can be observed that there are 13 bands within 0–200 Hz in **Figure 2(a)**. Besides the traditional plate modes, which are the in-plane modes (mainly the symmetric Lamb modes, such as modes *S2*) and the out-of-plane modes (mainly the antisymmetric Lamb modes, such as mode *A2*), lots of flat modes (such as modes *S1*, *A1*, *F2*), which are the resonant modes of the composite taper stubs, can be found. The bandgaps (one in-plane bandgap, one out-of-plane bandgap, and one complete bandgap), as a result of the coupling of the two kinds of modes mentioned above, appear. The in-plane bandgap (blue-dashed area: the frequency bands in which no in-plane modes) is due to the coupling between the in-plane modes (modes *S2*) and the corresponding flat modes (modes *S1*). It ranges from 53 to 93 Hz (between the fifth and eighth bands). The absolute bandwidth of it is 40 Hz. The out-of-plane bandgap (green-dashed area: the frequency bands in which no out-of-plane modes) is due to the coupling between the out-of-plane modes (mode *A2*) and the corresponding flat modes (mode *A1*). It ranges from 59 to 154 Hz (between the sixth and ninth bands), and the absolute bandwidth is 95 Hz; the complete bandgap (reddashed area: the frequency bands in which neither in-plane modes nor out-of-plane modes) is due to the overlap between the in-plane bandgap and the out-of-plane bandgap. It ranges from 59 to 93 Hz (between the sixth and eighth bands). The absolute bandwidth of it is 34 Hz. It can be observed that the location of the bandgap shifts into lower frequency (below 100 Hz), but the bandwidth is very narrow. As a comparison, we also calculated the band structures of the transition PC plate composed of double-sided composite taper stubs deposited on a homogeneous steel plate and the classical PC plate which was proposed by Assouar [9]. They are shown in **Figure 2(b)** and **(c)**, respectively. Their complete bandgaps (red-dashed areas) are both due to the overlap between the second in-plane bandgap and the first out-of-plane bandgap. It can be found clearly that the introduction of the

*Band structures of (a) the lower frequency complete bandgap metal-matrix embedded PC plate, (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-*

(hA = hB = 2.5 mm), dup = 9 mm, and dlow = 5 mm, respectively.

**Table 1.** *Material parameters in calculations.* *Photonic Crystals - A Glimpse of the Current Research Trends*

*TL* = 10 log(

**3.1 The lower frequency complete BG phononic crystals**

irreducible Brillouin zone.

**3. The model and the results**

*3.1.1 The model of the phononic crystals*

structure and its unit cell, respectively.

taper *B* are rubber and steel, respectively.

*3.1.2 The results of the phononic crystals*

**Material Mass density (kg/m3**

*Material parameters in calculations.*

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

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

where *αo* and *αi* are the output and input accelerations of the metal-matrix embedded phononic crystals, respectively. Finally, the transmission spectra can be

The lower frequency bandgap metal-matrix embedded phononic crystal [14] was composed of a square array of composite taper stubs on both sides of a twodimensional binary locally resonant PC plate which composes an array of rubber fillers embedded in the steel plate. **Figure 1(a)** and **(b)** shows part of the proposed

In the lower frequency bandgap metal-matrix embedded phononic crystals, the taper stub is composed of the *A* taper cap and the *B* which is located on the taper *A*. 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 taper stub are denoted by *h* (*hA* for taper *A* and *hB* for taper *B*) and *d* (the upper diameter of taper stub is denoted by *dup*, and the lower diameter is denoted by *dlow*), respectively. The material parameters used in the calculations are listed in **Table 1**. The taper *A* and the

Through the use of the finite element method, the systems described in **Figure 1** were studied numerically. The band structures and displacement vector

Steel 7800 210,000 0.29 Rubber 1300 0.1175 0.47

**) Young's modulus (106**

**N/m2**

**) Poisson's ratio**

obtained by changing the excitation frequency of the incident acceleration.

\_\_ α*o*

<sup>α</sup>*i*) (4)

**32**

**Table 1.**

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 = 5 mm (hA = hB = 2.5 mm), dup = 9 mm, and dlow = 5 mm, respectively.

It can be observed that there are 13 bands within 0–200 Hz in **Figure 2(a)**. Besides the traditional plate modes, which are the in-plane modes (mainly the symmetric Lamb modes, such as modes *S2*) and the out-of-plane modes (mainly the antisymmetric Lamb modes, such as mode *A2*), lots of flat modes (such as modes *S1*, *A1*, *F2*), which are the resonant modes of the composite taper stubs, can be found. The bandgaps (one in-plane bandgap, one out-of-plane bandgap, and one complete bandgap), as a result of the coupling of the two kinds of modes mentioned above, appear. The in-plane bandgap (blue-dashed area: the frequency bands in which no in-plane modes) is due to the coupling between the in-plane modes (modes *S2*) and the corresponding flat modes (modes *S1*). It ranges from 53 to 93 Hz (between the fifth and eighth bands). The absolute bandwidth of it is 40 Hz. The out-of-plane bandgap (green-dashed area: the frequency bands in which no out-of-plane modes) is due to the coupling between the out-of-plane modes (mode *A2*) and the corresponding flat modes (mode *A1*). It ranges from 59 to 154 Hz (between the sixth and ninth bands), and the absolute bandwidth is 95 Hz; the complete bandgap (reddashed area: the frequency bands in which neither in-plane modes nor out-of-plane modes) is due to the overlap between the in-plane bandgap and the out-of-plane bandgap. It ranges from 59 to 93 Hz (between the sixth and eighth bands). The absolute bandwidth of it is 34 Hz. It can be observed that the location of the bandgap shifts into lower frequency (below 100 Hz), but the bandwidth is very narrow.

As a comparison, we also calculated the band structures of the transition PC plate composed of double-sided composite taper stubs deposited on a homogeneous steel plate and the classical PC plate which was proposed by Assouar [9]. They are shown in **Figure 2(b)** and **(c)**, respectively. Their complete bandgaps (red-dashed areas) are both due to the overlap between the second in-plane bandgap and the first out-of-plane bandgap. It can be found clearly that the introduction of the

#### **Figure 2.**

*Band structures of (a) the lower frequency complete bandgap metal-matrix embedded PC plate, (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.*

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].
