*4.1.1 Forming mechanisms of the lower out-of-plane BGs*

**Figure 5** displays the magnitude of the total displacement vector of a unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals (**Figure 2(a)**), the transition structure (**Figure 2(b)**), and the classical structure (**Figure 2(c)**), respectively. They correspond to the upper and lower edge of the out-of-plane bandgap of each structure. The mode *A2* is an antisymmetric Lamb mode of the plate. The steel plate vibrates along the *z*-axis, while the stub remains stationary. In the frequencies, the antisymmetric Lamb mode will be activated, and the out-of-plane waves propagate through the PC plate in the antisymmetric Lamb mode. When the frequency of the out-of-plane waves is near the first nature frequency of the stub resonator, the resonant mode *A1* will be activated. The stub vibrates along the *z*-direction, and it gives a reacting force to the plate against the plate that vibrates along the *z*-direction. In that case, the out-of-plane waves are not capable of propagating through the PC plate. As a result, an out-of-plane bandgap is opened. Within the out-of-plane bandgap, the reacting force is still applied on the

#### **Figure 5.**

*The total displacement vector fields of the modes (resonant mode A1 and antisymmetric Lamb mode A2) (a) correspond to* **Figure 2(a)***, (b) correspond to* **Figure 2(b)***, and (c) correspond to* **Figure 2(c)***.*

**37**

**Figure 6.**

*matrix embedded phononic crystals.*

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

plate and prevents the propagation of the out-of-plane waves. The frequency of the out-of-plane waves deviates from the nature frequency of the mode *A1*, according to

*z* = *η*1*A*<sup>1</sup> + *η*2*A*<sup>2</sup> + ⋯ + *ηnAn* (5)

where *z* denotes the response of the plate and *ηn* denotes the modal participation factor of the mode *An*. The modal participation factor *η1* of the mode *A1* becomes small, leads the reacting force to becoming weak, and then disappears, and the antisymmetric Lamb mode *A2* is released again. As a result, the out-of-plane bandgap is closed. The formation mechanism of the bandgaps is shown in **Figure 6**. As the frequency of the out-of-plane waves deviates from the nature frequency of the resonator mode, the modal participation factors of it become small, lead the reacting force to becoming weak, and then disappear; the out-of-plane bandgap is closed. It can be concluded that the out-of-plane bandgap of the system is formed due to the coupling between the flat mode *A1* and the antisymmetric Lamb mode *A2* which is

The opening location of the out-of-plane bandgap is determined by the nature frequency of the resonant mode *A1*. The vibration process of the resonant mode *A1* can be understood as a mass-spring system whose frequency is determined by the formula

2*π* √

where *K1* is the spring stiffness and *M1* is the lump mass. For the classical structure, the rubber stub (denoted by *A*) acts as a spring, and the cap steel stub (denoted by *B*) acts as a mass (*M1 = MB*, where *MB* denotes the mass of stub *B*). For the lower frequency complete bandgap metal-matrix embedded phononic crystals, it can be found that the displacement fields are distributed in the whole stub and manifest an "analogous-rigid mode" of the whole stub, since the whole stub bodily moves along the *z*-axis with weak constrain, and the natural frequency is not zero. In this case, the rubber filler acts as a spring, and the whole stub acts as a mass; thus, the frequency is shifted to a lowest frequency range. It can be concluded that the out-of-plane bandgap is adjusted into lower frequency range by the rubber filler.

*The formation mechanism of the out-of-plane bandgaps of the lower frequency complete bandgap metal-*

\_\_\_ \_\_\_ *K*1 *M*<sup>1</sup>

(6)

the modal superposition principle which can be described as

established based on the modal superposition principle.

*f*<sup>1</sup> = \_\_\_1

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

3.37 compared with a classical phononic-crystal plate [18].

**4. Forming mechanisms of the BGs of the phononic crystals**

the first bandgap (in-plane and out-of-plane bandgaps), are extracted.

*4.1.1 Forming mechanisms of the lower out-of-plane BGs*

**4.1 Forming mechanisms of the lower frequency complete BGs**

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

In order to study the physical mechanism for the occurrence of the lower frequency complete bandgap in the metal-matrix embedded phononic crystals [14, 18], several specific resonance modes (mode *A1*, mode *S1*), which correspond to the lower edge of the first bandgap (out-of-plane and in-plane bandgap), and several specific traditional plate modes (mode *A2*, mode *S2*), which correspond to the upper edge of

**Figure 5** displays the magnitude of the total displacement vector of a unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals (**Figure 2(a)**), the transition structure (**Figure 2(b)**), and the classical structure (**Figure 2(c)**), respectively. They correspond to the upper and lower edge of the out-of-plane bandgap of each structure. The mode *A2* is an antisymmetric Lamb mode of the plate. The steel plate vibrates along the *z*-axis, while the stub remains stationary. In the frequencies, the antisymmetric Lamb mode will be activated, and the out-of-plane waves propagate through the PC plate in the antisymmetric Lamb mode. When the frequency of the out-of-plane waves is near the first nature frequency of the stub resonator, the resonant mode *A1* will be activated. The stub vibrates along the *z*-direction, and it gives a reacting force to the plate against the plate that vibrates along the *z*-direction. In that case, the out-of-plane waves are not capable of propagating through the PC plate. As a result, an out-of-plane bandgap is opened. Within the out-of-plane bandgap, the reacting force is still applied on the

*The total displacement vector fields of the modes (resonant mode A1 and antisymmetric Lamb mode A2) (a)* 

*correspond to* **Figure 2(a)***, (b) correspond to* **Figure 2(b)***, and (c) correspond to* **Figure 2(c)***.*

**36**

**Figure 5.**

plate and prevents the propagation of the out-of-plane waves. The frequency of the out-of-plane waves deviates from the nature frequency of the mode *A1*, according to the modal superposition principle which can be described as

$$z = \eta\_1 A\_1 + \eta\_2 A\_2 + \dots + \eta\_n A\_n \tag{5}$$

where *z* denotes the response of the plate and *ηn* denotes the modal participation factor of the mode *An*. The modal participation factor *η1* of the mode *A1* becomes small, leads the reacting force to becoming weak, and then disappears, and the antisymmetric Lamb mode *A2* is released again. As a result, the out-of-plane bandgap is closed. The formation mechanism of the bandgaps is shown in **Figure 6**. As the frequency of the out-of-plane waves deviates from the nature frequency of the resonator mode, the modal participation factors of it become small, lead the reacting force to becoming weak, and then disappear; the out-of-plane bandgap is closed. It can be concluded that the out-of-plane bandgap of the system is formed due to the coupling between the flat mode *A1* and the antisymmetric Lamb mode *A2* which is established based on the modal superposition principle.

The opening location of the out-of-plane bandgap is determined by the nature frequency of the resonant mode *A1*. The vibration process of the resonant mode *A1* can be understood as a mass-spring system whose frequency is determined by the formula

$$f\_1 = \frac{1}{2\pi} \sqrt{\frac{K\_1}{M\_1}}\tag{6}$$

where *K1* is the spring stiffness and *M1* is the lump mass. For the classical structure, the rubber stub (denoted by *A*) acts as a spring, and the cap steel stub (denoted by *B*) acts as a mass (*M1 = MB*, where *MB* denotes the mass of stub *B*). For the lower frequency complete bandgap metal-matrix embedded phononic crystals, it can be found that the displacement fields are distributed in the whole stub and manifest an "analogous-rigid mode" of the whole stub, since the whole stub bodily moves along the *z*-axis with weak constrain, and the natural frequency is not zero. In this case, the rubber filler acts as a spring, and the whole stub acts as a mass; thus, the frequency is shifted to a lowest frequency range. It can be concluded that the out-of-plane bandgap is adjusted into lower frequency range by the rubber filler.

#### **Figure 6.**

*The formation mechanism of the out-of-plane bandgaps of the lower frequency complete bandgap metalmatrix embedded phononic crystals.*

**Figure 7.**

*The total displacement vector fields of the modes (resonant mode S1, symmetric Lamb mode S2) (a) correspond to Figure 2(a), (b) correspond to Figure 2(b), and (c) correspond to Figure 2(c).*

### *4.1.2 Forming mechanisms of the lower in-plane BGs*

**Figure 7** displays the magnitudes of the total displacement vectors of a unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals (**Figure 7(a)**), the transition structure (**Figure 7(b)**), and the classical structure (**Figure 7(c)**), respectively. They correspond to the upper and lower edges of the first in-plane bandgap of each structure. The mode *S2* is a symmetric Lamb mode of the plate. The steel plate vibrates along the *xy*-plane, while the stubs "swing" in the opposite direction. In the frequencies, the symmetric Lamb mode *S2* will be activated, and the in-plane waves propagate through the PC plate in the symmetric Lamb mode. When the frequency of the in-plane waves is near the first nature frequency of stub resonator, the resonant mode *S1* (flat mode) will be activated. The stub "swings" along a plane which is vertical to the *xy*-plane, and it gives a reacting force to the plate to prevent the plate to vibrate along the *xy*plane. In that case, the in-plane waves are not capable of propagating through the PC plate. As a result, an in-plane bandgap is opened. Within the in-plane bandgap, the reacting force is still applied on the plate and prevents in-plane waves that propagate. The frequency of the in-plane waves deviates from the nature frequency of the resonant mode *S1*, according to the modal superposition principle. Which can be described as

$$\mathbf{x}\mathbf{y} = \eta\_1 \mathbf{S}\_1 + \eta\_2 \mathbf{S}\_2 + \dots + \eta\_n \mathbf{S}\_n \tag{7}$$

**39**

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

frequency range.

*f*<sup>2</sup> = \_\_\_1

*4.1.3 Forming mechanisms of the lower complete BGs*

quency, and finally a lowest complete bandgap is formed [14].

**4.2 Forming mechanisms of the broad complete BGs**

(in-plane and out-of-plane bandgaps), are extracted.

*4.2.1 Forming mechanisms of the broad out-of-plane BGs*

2*π* √

We can conclude from the above investigations that the resonator in the metalmatrix embedded phononic crystals can be considered as a spring-mass system, as shown in **Figures 6** and **7**. The spring-mass system includes two subsystems. The first one is the *K1-M1* subsystem (also refer to **Figure 6**). Its local resonance mode is coupled with the antisymmetric Lamb mode of the plate according to the modal superposition principle, and then the out-of-plane bandgap is generated. The other one is the *K2-M2* subsystem (also **Figure 7**). Its local resonance mode is coupled with the symmetric Lamb mode of the plate according to the modal superposition principle and is responsible for the formation of the in-plane bandgaps. The opening locations of the in-plane and out-of-plane BGs depend on the natural frequencies of the two subsystems.

As for the classical and the transition metal-matrix embedded phononic crystals, the steel stub (stub *B*) acts as a mass, and the rubber stub (stub *A*) acts as a spring, which leads to the coupling between the *K1-M1* subsystem and *K2-M2* subsystem. Therefore, it is difficult to adjust the in-plane and out-of-plane bandgaps separately. In the lower frequency bandgap metal-matrix embedded phononic crystals, the rubber filler acts as the stiffness *K1*, the stub *A* acts as the stiffness *K2*, the whole stub acts as the mass *M1*, and the stub *B* acts as the mass *M2*. As a result, the coupling between the *K1-M1* subsystem and the *K2-M2* subsystem can be decoupled. Moreover, the mass *M1* is magnified due to the "analogous-rigid mode" of the whole stub. Therefore, the out-of-plane bandgap can be adjusted into the lowest frequency range. Additionally, the stiffness *K2* can be reduced by introducing the taper stub, and thus the in-plane bandgaps can be adjusted into the lowest frequency range. As a result, the two bandgaps can be overlapped with each other in the lowest fre-

In order to study the physical mechanism for the occurrence of a broad complete bandgap in the metal-matrix embedded phononic crystals, several specific resonance modes (modes *A*1 and *S*1), which correspond to the lower edge of the first bandgap (out-of-plane and in-plane bandgap), and several specific traditional plate modes (mode *A*2 and *S*2), which correspond to the upper edge of the first bandgap

**Figure 8** displays the magnitude of the total displacement vector of a unit cell of the broad complete bandgap metal-matrix embedded phononic crystals (**Figure 8(a)**), the transition PC plate (**Figure 8(b))**, and the classical PC plate (**Figure 8(c)**), respectively. They correspond to the upper and lower edge of the out-of-plane bandgap of each structure. The mode *A*2 is an antisymmetric Lamb mode of the plate. The steel plate vibrates along the *z*-axis, while the stub remains

where *K2* is the tensional stiffness of the spring and *M2* is the lump mass. For the three PC plates, the rubber stub *A* mainly acts as a spring, and the cap steel stub *B* acts as a mass (*M2 = MB*). As the stiffness of the taper rubber stub is weaker than that of the cylinder stub, the location of the in-plane bandgap is adjusted into lower

\_\_\_ \_\_\_ *K*2 *M*<sup>2</sup>

(8)

where *xy* denotes the response of the plate and *ηn* denotes the modal participation factor of the mode *Sn*. The modal participation factor *η1* of the mode *S1* becomes small, leads the reacting force to becoming weak, and then disappears, and then the symmetric Lamb mode *S2* is released again. As a result, an in-plane bandgap is closed. It can be concluded that the first in-plane bandgap of the system is formed due to the coupling between the flat mode *S1* and the symmetric Lamb mode *S2* which is established according to the modal superposition principle. The opening location is determined by the nature frequency of the resonant mode *S1*. The resonant mode *S1* can also be understood as a "mass-spring" system whose frequency is determined by the formula

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

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

*4.1.2 Forming mechanisms of the lower in-plane BGs*

*to Figure 2(a), (b) correspond to Figure 2(b), and (c) correspond to Figure 2(c).*

**Figure 7** displays the magnitudes of the total displacement vectors of a unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals (**Figure 7(a)**), the transition structure (**Figure 7(b)**), and the classical structure (**Figure 7(c)**), respectively. They correspond to the upper and lower edges of the first in-plane bandgap of each structure. The mode *S2* is a symmetric Lamb mode of the plate. The steel plate vibrates along the *xy*-plane, while the stubs "swing" in the opposite direction. In the frequencies, the symmetric Lamb mode *S2* will be activated, and the in-plane waves propagate through the PC plate in the symmetric Lamb mode. When the frequency of the in-plane waves is near the first nature frequency of stub resonator, the resonant mode *S1* (flat mode) will be activated. The stub "swings" along a plane which is vertical to the *xy*-plane, and it gives a reacting force to the plate to prevent the plate to vibrate along the *xy*plane. In that case, the in-plane waves are not capable of propagating through the PC plate. As a result, an in-plane bandgap is opened. Within the in-plane bandgap, the reacting force is still applied on the plate and prevents in-plane waves that propagate. The frequency of the in-plane waves deviates from the nature frequency of the resonant mode *S1*, according to the modal superposition principle.

*The total displacement vector fields of the modes (resonant mode S1, symmetric Lamb mode S2) (a) correspond* 

*xy* = *η*1*S*<sup>1</sup> + *η*2*S*<sup>2</sup> + ⋯ + *η<sup>n</sup> Sn* (7)

where *xy* denotes the response of the plate and *ηn* denotes the modal participation factor of the mode *Sn*. The modal participation factor *η1* of the mode *S1* becomes small, leads the reacting force to becoming weak, and then disappears, and then the symmetric Lamb mode *S2* is released again. As a result, an in-plane bandgap is closed. It can be concluded that the first in-plane bandgap of the system is formed due to the coupling between the flat mode *S1* and the symmetric Lamb mode *S2* which is established according to the modal superposition principle. The opening location is determined by the nature frequency of the resonant mode *S1*. The resonant mode *S1* can also be understood as a "mass-spring" system whose frequency is determined by the formula

**38**

Which can be described as

**Figure 7.**

$$f\_2 = \frac{1}{2\pi} \sqrt{\frac{K\_2}{M\_2}}\tag{8}$$

where *K2* is the tensional stiffness of the spring and *M2* is the lump mass. For the three PC plates, the rubber stub *A* mainly acts as a spring, and the cap steel stub *B* acts as a mass (*M2 = MB*). As the stiffness of the taper rubber stub is weaker than that of the cylinder stub, the location of the in-plane bandgap is adjusted into lower frequency range.

#### *4.1.3 Forming mechanisms of the lower complete BGs*

We can conclude from the above investigations that the resonator in the metalmatrix embedded phononic crystals can be considered as a spring-mass system, as shown in **Figures 6** and **7**. The spring-mass system includes two subsystems. The first one is the *K1-M1* subsystem (also refer to **Figure 6**). Its local resonance mode is coupled with the antisymmetric Lamb mode of the plate according to the modal superposition principle, and then the out-of-plane bandgap is generated. The other one is the *K2-M2* subsystem (also **Figure 7**). Its local resonance mode is coupled with the symmetric Lamb mode of the plate according to the modal superposition principle and is responsible for the formation of the in-plane bandgaps. The opening locations of the in-plane and out-of-plane BGs depend on the natural frequencies of the two subsystems.

As for the classical and the transition metal-matrix embedded phononic crystals, the steel stub (stub *B*) acts as a mass, and the rubber stub (stub *A*) acts as a spring, which leads to the coupling between the *K1-M1* subsystem and *K2-M2* subsystem. Therefore, it is difficult to adjust the in-plane and out-of-plane bandgaps separately. In the lower frequency bandgap metal-matrix embedded phononic crystals, the rubber filler acts as the stiffness *K1*, the stub *A* acts as the stiffness *K2*, the whole stub acts as the mass *M1*, and the stub *B* acts as the mass *M2*. As a result, the coupling between the *K1-M1* subsystem and the *K2-M2* subsystem can be decoupled. Moreover, the mass *M1* is magnified due to the "analogous-rigid mode" of the whole stub. Therefore, the out-of-plane bandgap can be adjusted into the lowest frequency range. Additionally, the stiffness *K2* can be reduced by introducing the taper stub, and thus the in-plane bandgaps can be adjusted into the lowest frequency range. As a result, the two bandgaps can be overlapped with each other in the lowest frequency, and finally a lowest complete bandgap is formed [14].

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

In order to study the physical mechanism for the occurrence of a broad complete bandgap in the metal-matrix embedded phononic crystals, several specific resonance modes (modes *A*1 and *S*1), which correspond to the lower edge of the first bandgap (out-of-plane and in-plane bandgap), and several specific traditional plate modes (mode *A*2 and *S*2), which correspond to the upper edge of the first bandgap (in-plane and out-of-plane bandgaps), are extracted.

#### *4.2.1 Forming mechanisms of the broad out-of-plane BGs*

**Figure 8** displays the magnitude of the total displacement vector of a unit cell of the broad complete bandgap metal-matrix embedded phononic crystals (**Figure 8(a)**), the transition PC plate (**Figure 8(b))**, and the classical PC plate (**Figure 8(c)**), respectively. They correspond to the upper and lower edge of the out-of-plane bandgap of each structure. The mode *A*2 is an antisymmetric Lamb mode of the plate. The steel plate vibrates along the *z*-axis, while the stub remains

#### **Figure 8.**

*The total displacement vector fields of the modes (resonant mode A1 and antisymmetric Lamb mode A2) (a) correspond to Figure 4(b), (b) correspond to Figure 4(b), and (c) correspond to Figure 4(c).*

stationary. With regard to the frequencies, the antisymmetric Lamb mode will be activated, and the out-of-plane waves propagate through the phononic-crystal plate in the antisymmetric Lamb mode. When the frequency of the out-of-plane waves is near the first nature frequency of the stub resonator, the resonant mode *A*1 will be activated. The stub vibrates along the *z*-direction and produces a reacting force for the plate, and the plate vibrates along the *z*-direction. In this case, the out-ofplane waves are not capable of propagating through the phononic-crystal plate. As a result, an out-of-plane bandgap is created. The bandwidth is determined by the coupling between the resonance mode *A*1 and the traditional plate *A*2 mode.

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

$$F = k\_o \varepsilon\_T \tag{9}$$

**41**

**Figure 10.**

*embedded phononic crystals.*

**Figure 9.**

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

along the *z*-direction (see **Figure 9(a)** and **(c)**).

The whole stub vibrates in the *z*-direction (out-of-plane) and generates a reacting force. *F*1 and *F*2 respond to the broad complete bandgap metal-matrix embedded phononic crystals and the transition PC plate, respectively. *F*1 = *k*0.*s*1 = *k*RT.*s*1, where *k*RT is the transverse stiffness of the rubber filler and *s*1 is the vibration amplitude of the whole single "hard" stub; *F*2 = *k*0.*s*2 = *k*RT.*s*2, where *s*2 is the vibration amplitude of the whole composite stub through rubber filler to the plate against the plate vibrates

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

In this case, the rubber filler acts as a spring, and the whole stub acts as a mass. The broad complete bandgap metal-matrix embedded phononic crystals

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

where sT is the vibration amplitude of the stub and *k*o is the spring stiffness.

For the classical structure, the formation mechanism of the out-of-plane bandgap is shown in **Figure 9(b)**. The hard stub (rigid body) vibrates along the z-direction and generates a reacting force through the soft stub (flexible body) to the plate against the plate vibrating along the *z*-direction. The soft stub acts as a spring, while the hard stub acts as a mass (ko = ksc, *m*o = *m*h, where *m*h denotes the mass of the hard stub and ksc denotes the compression stiffness of the soft stub) (see **Figure 10(b)**).

For both the broad complete bandgap metal-matrix embedded phononic crystals and the transition PC plate, it can be found that the displacement fields are distributed in the whole stub (see **Figure 8(a)** and **(b)**), respectively. This results in an "out-of-plane analogous-rigid mode" because the whole stub vibrates along the *z*-axis (out-of-plane) with a weak constraint, while the frequency is non-zero. The displacement fields of its eigenmodes are distributed throughout the whole stub. This means that the whole stub body moves along the z-direction 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 "rigid mode" and call these types of vibration modes for the whole stub the "out-of-plane analogousrigid mode." The formation mechanisms for the out-of-plane bandgap of the two structures are shown in **Figure 9(a)** and **(c)**, respectively.

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

#### **Figure 9.**

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

stationary. With regard to the frequencies, the antisymmetric Lamb mode will be activated, and the out-of-plane waves propagate through the phononic-crystal plate in the antisymmetric Lamb mode. When the frequency of the out-of-plane waves is near the first nature frequency of the stub resonator, the resonant mode *A*1 will be activated. The stub vibrates along the *z*-direction and produces a reacting force for the plate, and the plate vibrates along the *z*-direction. In this case, the out-ofplane waves are not capable of propagating through the phononic-crystal plate. As a result, an out-of-plane bandgap is created. The bandwidth is determined by the coupling between the resonance mode *A*1 and the traditional plate *A*2 mode.

*The total displacement vector fields of the modes (resonant mode A1 and antisymmetric Lamb mode A2) (a)* 

*correspond to Figure 4(b), (b) correspond to Figure 4(b), and (c) correspond to Figure 4(c).*

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

*F* = *ko sT* (9)

where sT is the vibration amplitude of the stub and *k*o is the spring stiffness. For the classical structure, the formation mechanism of the out-of-plane bandgap is shown in **Figure 9(b)**. The hard stub (rigid body) vibrates along the z-direction and generates a reacting force through the soft stub (flexible body) to the plate against the plate vibrating along the *z*-direction. The soft stub acts as a spring, while the hard stub acts as a mass (ko = ksc, *m*o = *m*h, where *m*h denotes the mass of the hard stub and ksc denotes the compression stiffness of the soft stub) (see **Figure 10(b)**). For both the broad complete bandgap metal-matrix embedded phononic crystals

and the transition PC plate, it can be found that the displacement fields are distributed in the whole stub (see **Figure 8(a)** and **(b)**), respectively. This results in an "out-of-plane analogous-rigid mode" because the whole stub vibrates along the *z*-axis (out-of-plane) with a weak constraint, while the frequency is non-zero. The displacement fields of its eigenmodes are distributed throughout the whole stub. This means that the whole stub body moves along the z-direction 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 "rigid mode" and call these types of vibration modes for the whole stub the "out-of-plane analogousrigid mode." The formation mechanisms for the out-of-plane bandgap of the two

structures are shown in **Figure 9(a)** and **(c)**, respectively.

*A*2 mode is determined by the formula

**Figure 8.**

**40**

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

The whole stub vibrates in the *z*-direction (out-of-plane) and generates a reacting force. *F*1 and *F*2 respond to the broad complete bandgap metal-matrix embedded phononic crystals and the transition PC plate, respectively. *F*1 = *k*0.*s*1 = *k*RT.*s*1, where *k*RT is the transverse stiffness of the rubber filler and *s*1 is the vibration amplitude of the whole single "hard" stub; *F*2 = *k*0.*s*2 = *k*RT.*s*2, where *s*2 is the vibration amplitude of the whole composite stub through rubber filler to the plate against the plate vibrates along the *z*-direction (see **Figure 9(a)** and **(c)**).

In this case, the rubber filler acts as a spring, and the whole stub acts as a mass. The broad complete bandgap metal-matrix embedded phononic crystals

#### **Figure 10.**

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

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