*4.2.2 Forming mechanisms of the broad in-plane BGs*

**Figure 11** displays the magnitudes of the total displacement vectors of a unit cell of the broad complete bandgap metal-matrix embedded phononic crystals, the transition PC plate, and the classical PC plate. The figures correspond to the upper and lower edges of the first in-plane bandgap of each structure. The mode *S*<sup>2</sup> is a symmetric Lamb mode of the plate. The steel plate vibrates along the *xy*-plane (in-plane), while the stubs swing in the opposite direction. With respect to the frequencies, the symmetric Lamb mode will be activated, and the in-plane waves can propagate through the phononic-crystal plate in the symmetric Lamb mode. When the frequency of the in-plane waves approaches the first natural frequency of the stub resonator, the resonant mode *S*1 will be activated. The stub vibrates in the *yz*-plane and produces a reacting force to the plate while vibrating in the *xy*-direction. In that case, the in-plane waves are not able to propagate through the PCs. As a result, an in-plane bandgap opens up. The bandwidth is determined by the coupling between the resonance mode *S*1 and the traditional plate mode *S*2.

#### **Figure 11.**

*The total displacement vector fields for the modes (resonant mode S1, symmetric Lamb mode S2) (a) correspond to Figure 4(a), (b) correspond to Figure 4(b), and (c) correspond to Figure 4(c).*

**43**

**Figure 12.**

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

mode *S*2 is determined by the formula

the spring.

The coupling strength between the resonance mode *S*1 and the traditional plate

*F* = *kisL* (10)

where sL is the vibration amplitude of the stub and *k*i is the tensional stiffness of

For both the transition and the classical phononic-crystal plates, the formation mechanism of the in-plane bandgap is shown in **Figure 12(a)** and **(b)**, respectively. It can be found that the stubs "swing" in the *xy*-plane and produce a reacting force (*F*2 = *k*i.*s*L = *k*ST.*s*L, where *k*ST is the transverse stiffness of the soft stub and *s*L is the vibration amplitude of the whole stub) through the soft stub (flexible body) to the plate against the plate vibrating along the *xy-*plane. The soft stub (flexible) acts as a spring (*k*i = *k*s), while the hard stub (rigid body) acts as a mass (*m*i = *m*h), as shown in **Figure 13(a)** and **(b)**, respectively. Hence, both location and bandwidth

For the broad complete bandgap metal-matrix embedded phononic crystals, it can be found that the displacement fields are distributed across the entire stub (**Figure 11(a)**) to cause an "in-plane analogous-rigid mode" because the whole stub vibrates along the *xy-*plane (in-plane) with a weak constraint, while the frequency is non-zero. It can be observed that the displacement fields of its eigenmodes are distributed throughout the whole stub. This means that the whole stub body moves in the *xy*-plane like a rigid body moves in rigid mode. However, the natural frequency is not 0, and the whole stub is constrained by the rubber filler. Therefore, we refer to the concept of rigid mode and call this type of vibration mode of a whole stub the "in-plane analogous-rigid mode." The formation mechanism of the bandgap is shown in **Figure 12(c)**. The whole stub vibrates in the *xy-*plane and produces a reacting force (*F*1 = *k*i.*s*L = *k*RL.*s*L, where *k*RL is the longitudinal stiffness of the rubber filler and *s*L is the vibration amplitude of the whole stub) through the rubber

In this case, the "rubber filler" acts as a spring, where the longitudinal stiffness of the rubber filler acts as the spring, while the whole stub acts as a mass, where

*Formation mechanism process of the bandgap for the in-plane bandgap of (a) the "transition" PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.*

of the first in-plane bandgap in these structures are the same.

filler to the plate against the plate vibrating in the *xy-*plane.

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stiffness of the out-of-plane analogous-rigid mode of the stub.

between the resonance mode *S*1 and the traditional plate mode *S*2.

*The total displacement vector fields for the modes (resonant mode S1, symmetric Lamb mode S2) (a) correspond to Figure 4(a), (b) correspond to Figure 4(b), and (c) correspond to Figure 4(c).*

*4.2.2 Forming mechanisms of the broad in-plane BGs*

includes *m*<sup>o</sup> *= 2m*h and *m*<sup>o</sup> *= 2m*<sup>s</sup> *+ 2m*h, wher*e m*s denotes the mass of soft stub and *m*h denotes the mass of hard stub. It is shown in **Figures 9(c)** and **10(c)** that the frequency is shifted to the lowest frequency range. Compared to the transition PC plate, where the soft stub contacts the rubber filler such that it represents a flexible constraint to the rubber filler (see **Figure 9(a)**), the hard stub contacts the rubber filler in the broad complete bandgap metal-matrix embedded phononic crystals to produce a rigid constraint for the rubber filler (see **Figure 9(c)**). This causes the spring stiffness *k*0 (*k*0 = *k*RT*—*the longitudinal stiffness of the rubber filler) to increase, while the lump mass becomes smaller (*2m*<sup>s</sup> *+ 2m*h > *2m*h). This, in turn, not only causes the opening location of the out-of-plane bandgap to shift to higher frequencies but also makes the force *F*1 larger than the force *F*2. As a result, the outof-plane bandwidth becomes wider. We conclude that, after introducing the rubber filler, an out-of-plane analogous-rigid mode of the stub was produced, which can reduce the location of the out-of-plane bandgap. Hence, the introduction of a single "hard" stub increases the bandwidth of the out-of-plane bandgap by enhancing the

**Figure 11** displays the magnitudes of the total displacement vectors of a unit cell of the broad complete bandgap metal-matrix embedded phononic crystals, the transition PC plate, and the classical PC plate. The figures correspond to the upper and lower edges of the first in-plane bandgap of each structure. The mode *S*<sup>2</sup> is a symmetric Lamb mode of the plate. The steel plate vibrates along the *xy*-plane (in-plane), while the stubs swing in the opposite direction. With respect to the frequencies, the symmetric Lamb mode will be activated, and the in-plane waves can propagate through the phononic-crystal plate in the symmetric Lamb mode. When the frequency of the in-plane waves approaches the first natural frequency of the stub resonator, the resonant mode *S*1 will be activated. The stub vibrates in the *yz*-plane and produces a reacting force to the plate while vibrating in the *xy*-direction. In that case, the in-plane waves are not able to propagate through the PCs. As a result, an in-plane bandgap opens up. The bandwidth is determined by the coupling

**42**

**Figure 11.**

The coupling strength between the resonance mode *S*1 and the traditional plate mode *S*2 is determined by the formula

$$F = k\_i \mathfrak{s}\_L \tag{10}$$

where sL is the vibration amplitude of the stub and *k*i is the tensional stiffness of the spring.

For both the transition and the classical phononic-crystal plates, the formation mechanism of the in-plane bandgap is shown in **Figure 12(a)** and **(b)**, respectively.

It can be found that the stubs "swing" in the *xy*-plane and produce a reacting force (*F*2 = *k*i.*s*L = *k*ST.*s*L, where *k*ST is the transverse stiffness of the soft stub and *s*L is the vibration amplitude of the whole stub) through the soft stub (flexible body) to the plate against the plate vibrating along the *xy-*plane. The soft stub (flexible) acts as a spring (*k*i = *k*s), while the hard stub (rigid body) acts as a mass (*m*i = *m*h), as shown in **Figure 13(a)** and **(b)**, respectively. Hence, both location and bandwidth of the first in-plane bandgap in these structures are the same.

For the broad complete bandgap metal-matrix embedded phononic crystals, it can be found that the displacement fields are distributed across the entire stub (**Figure 11(a)**) to cause an "in-plane analogous-rigid mode" because the whole stub vibrates along the *xy-*plane (in-plane) with a weak constraint, while the frequency is non-zero. It can be observed that the displacement fields of its eigenmodes are distributed throughout the whole stub. This means that the whole stub body moves in the *xy*-plane like a rigid body moves in rigid mode. However, the natural frequency is not 0, and the whole stub is constrained by the rubber filler. Therefore, we refer to the concept of rigid mode and call this type of vibration mode of a whole stub the "in-plane analogous-rigid mode." The formation mechanism of the bandgap is shown in **Figure 12(c)**. The whole stub vibrates in the *xy-*plane and produces a reacting force (*F*1 = *k*i.*s*L = *k*RL.*s*L, where *k*RL is the longitudinal stiffness of the rubber filler and *s*L is the vibration amplitude of the whole stub) through the rubber filler to the plate against the plate vibrating in the *xy-*plane.

In this case, the "rubber filler" acts as a spring, where the longitudinal stiffness of the rubber filler acts as the spring, while the whole stub acts as a mass, where

#### **Figure 12.**

*Formation mechanism process of the bandgap for the in-plane bandgap of (a) the "transition" PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.*

#### **Figure 13.**

*The equivalent theoretical model of the resonator for formation mechanism of the in-plane bandgap of (a) the "transition" PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.*

*m*<sup>i</sup> *= m*h, and *m*h denotes the mass of hard stub (see **Figure 13(c)**). Compared to the transition PC plate, in which the soft stub contacts the rubber filler such that it gives a flexible constraint to the rubber filler, this leads to the soft stub acting as a spring (see **Figure 13(c)**). The hard stub contacts the rubber filler in the broad complete bandgap metal-matrix embedded phononic crystals, which creates a rigid constraint to the rubber filler that causes the rubber filler to act as a spring (see **Figure 13(c)**). This causes the spring stiffness *k*i to increase in the broad complete bandgap metalmatrix embedded phononic crystals such that the opening location of the in-plane bandgap is shifted toward higher frequencies. However, it also makes the force *F*1 larger than force *F*2 and causes the in-plane bandwidth to become broader. We conclude that, after introducing the rubber filler and the single "hard" stub simultaneously, an "in-plane analogous-rigid mode" of the stub is produced, which can increase the bandwidth of the in-plane bandgap.

### *4.2.3 Forming mechanisms of the broad complete BGs*

When the broad out-of-plane bandgaps and broad in-plane bandgaps, which were simultaneously increased by the single "hard" stub, overlap, a wider complete bandgap, in which the out-of-plane and in-plane Lamb waves are prohibited, is created.

It can be concluded that the bandwidth of the bandgap is determined by the resonator mode. After introducing the rubber filler and the single "hard" stub simultaneously, two new resonator modes are produced. One resonator mode is the in-plane analogous-rigid mode, in which the whole stub vibrates along the in-plane plate. The other resonator mode is the out-of-plane analogous-rigid mode, in which the stub vibrates along the out-of-plane plate. They both increase the in-plane and out-ofplane bandgaps, respectively. Two increased bandgaps overlap to produce a broad complete bandgap, in which both the out-of-plane and in-plane Lamb waves are

**45**

**Figure 14.**

*e = 1 mm, and a = 10 mm, respectively.*

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

**5. The effect of the stubs on the BGs**

stub height on the first complete bandgap.

steel-stub height increases.

prohibited. The rubber filler shifts the two kinds of bandgaps simultaneously toward low frequency such that a broad complete bandgap for low frequencies is obtained.

In order to investigate the effect of the stubs on the complete bandgaps of the metal-matrix embedded phononic crystals [14, 18], we studied the influence of the

**Figure 14** displays the evolution of the first complete bandgap as a function of the steel-stub height, hS. We can find that, with the increase of the steel-stub height, both the lower and upper edges of the first complete shift to a lower frequency range firstly and then move to higher frequency range. For example, when the steel-stub height is less than or equal to 3 mm, with the increase of the steel-stub height, the lower edge shifts to lower frequencies, but when the rubber stub height is larger than 3 mm, the lower edge frequency shifts to higher frequencies as the

In order to investigate the effect of single "hard" stubs on the complete phononic

We find that, with the increase of the steel-stub height, the location of the first

bandgaps of the novel metal-matrix PCs, we studied the effect of the steel-stub height on the first complete bandgap. **Figure 15** displays the evolution of the first

complete bandgap shifts to a lower frequency range before it shifts to a higher frequency range. Furthermore, the bandwidth of the bandgaps becomes broad. For example, when the steel-stub height is below or equal to 3.5 mm, after increasing the steel-stub height, the location of the bandgaps shifts to lower frequencies. However, when the steel-stub height exceeds 3.5 mm, the location of the bandgaps

*The evolution of the first complete BG in the lower frequency complete bandgap metal-matrix embedded phononic crystals as a function of the steel-stub height with D = 8 mm, dup = 9 mm, dup = 5 mm, h = 5 mm,* 

**5.1 The effect of the stubs on the lower frequency complete BGs**

**5.2 The effect of the stubs on the broad complete BGs**

complete bandgap as a function of the steel-stub height *h*.

shifts to higher frequencies as the steel-stub height increases.

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increase the bandwidth of the in-plane bandgap.

*4.2.3 Forming mechanisms of the broad complete BGs*

*m*<sup>i</sup> *= m*h, and *m*h denotes the mass of hard stub (see **Figure 13(c)**). Compared to the transition PC plate, in which the soft stub contacts the rubber filler such that it gives a flexible constraint to the rubber filler, this leads to the soft stub acting as a spring (see **Figure 13(c)**). The hard stub contacts the rubber filler in the broad complete bandgap metal-matrix embedded phononic crystals, which creates a rigid constraint to the rubber filler that causes the rubber filler to act as a spring (see **Figure 13(c)**). This causes the spring stiffness *k*i to increase in the broad complete bandgap metalmatrix embedded phononic crystals such that the opening location of the in-plane bandgap is shifted toward higher frequencies. However, it also makes the force *F*1 larger than force *F*2 and causes the in-plane bandwidth to become broader. We conclude that, after introducing the rubber filler and the single "hard" stub simultaneously, an "in-plane analogous-rigid mode" of the stub is produced, which can

*The equivalent theoretical model of the resonator for formation mechanism of the in-plane bandgap of (a) the "transition" PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded* 

When the broad out-of-plane bandgaps and broad in-plane bandgaps, which were simultaneously increased by the single "hard" stub, overlap, a wider complete bandgap, in which the out-of-plane and in-plane Lamb waves are prohibited, is created. It can be concluded that the bandwidth of the bandgap is determined by the resonator mode. After introducing the rubber filler and the single "hard" stub simultaneously, two new resonator modes are produced. One resonator mode is the in-plane analogous-rigid mode, in which the whole stub vibrates along the in-plane plate. The other resonator mode is the out-of-plane analogous-rigid mode, in which the stub vibrates along the out-of-plane plate. They both increase the in-plane and out-ofplane bandgaps, respectively. Two increased bandgaps overlap to produce a broad complete bandgap, in which both the out-of-plane and in-plane Lamb waves are

**44**

**Figure 13.**

*phononic crystals.*

prohibited. The rubber filler shifts the two kinds of bandgaps simultaneously toward low frequency such that a broad complete bandgap for low frequencies is obtained.
