Design of a Functionally Graded Material Phonon Crystal Plate and Its Application in a Bridge

Abstract

In order to alleviate the structural vibrations induced by traffic loads, in this paper, a phonon crystal plate with functionally graded materials is designed based on local resonance theory. The vibration damping performance of the phonon crystal plate is studied via finite element numerical simulation and the band gap is verified via vibration transmission response analysis. Finally, the engineering application mode is simulated to make it have practical engineering application value. The results show that the phonon crystal plate has two complete bandgaps within 0~150 Hz, the initial bandgap frequency is 0.00 Hz, the cut-off frequency is 128.32 Hz, and the internal ratio of 0~100 Hz is 94.13%, which can effectively reduce the structural vibration caused by traffic loads. Finally, stress analysis of the phonon crystal plate is carried out. The results show that phonon crystals of functionally graded materials can reduce stress concentration through adjusting the band gap. The phonon crystal plate designed in this paper can effectively suppress the structural vibration caused by traffic loads, provides a new method for the vibration reduction of traffic infrastructure, and can be applied to the vibration reduction of bridges and their auxiliary facilities.

1. Introduction

Expansion joints are currently installed in the design of bridge structures in order to avoid differences in temperature changes that cause the pavement structure to be affected by thermal expansion and contraction, while playing a role in moderating the longitudinal impact [1,2]. However, the setting of expansion joints also reduces the integrity of the bridge structure and the pavement, and when traffic loads such as cars and trains pass by, it will cause the bridge structure to produce violent vibrations and noise, which will be transmitted to the surrounding area through the ground and other appurtenances to bring a great impact on people’s lives and also lead to the destruction of the bridge structure [3,4,5]. Therefore, vibration reduction of bridge structures has become an important issue for scholars to study at present.

In recent years, phononic crystals (PCs) have been widely used in engineering structures for vibration and noise reduction due to their unique elastic wave bandgap (BG) properties [6,7]. The forbidden band formation mechanism can be divided into two modes, Bragg scattering and local resonance. The Bragg-scattering-type PC forbidden band formation mechanism is the anti-phase superposition between elastic waves [8], so it requires the lattice constant to be in the same order of magnitude as the wavelength. Local resonant PCs [9,10,11], with their unique structural design, have band gap frequencies that are several orders of magnitude lower than Bragg band gap frequencies for the same lattice constants [12], extending the application range of phonon crystals [13].

Wu et al. [14] designed a two-basis locally resonant PC and applied it to the interior of a car for vibration and noise reduction, resulting in a 15 dB reduction in overall body vibration level and a 5 dB reduction in interior noise.

Functional gradient materials [15] are a new type of composite material with a continuous gradient change in composition and structure, which can achieve functional and performance changes with the internal location of the material.

Wu et al. [16] studied the energy band structure of one-dimensional PCs containing functional gradient materials using the finite element method and analyzed the influencing factors, and the results showed that the BG frequency of PCs containing functional gradient materials is higher than that of PCs of conventional materials.

Wei et al. [17] designed two-dimensional PCs containing functional gradient materials with three different material combinations and investigated their BG characteristics, and the results showed that the functional gradient materials had the most significant effect on the BG of solid–solid PCs.

In summary, PCs have been applied in engineering structures due to their excellent vibration and noise reduction properties in established research, but research on structural vibrations caused by traffic loads is still relatively rare. Functional gradient materials have received attention in PC research due to their properties of adjustable BG and the advantage of reducing stress concentration, but as a composite material, their effect on the forbidden band of PCs has not been sufficiently studied due to the diversity of their material and physical parameters. Based on this paper, we combine the properties of functional gradient materials and PCs to design a PC plate with functional gradient materials and investigate its BG characteristics in order to provide a new method for vibration reduction in traffic engineering.

2. Numerical Model and Method

2.1. Phonon Crystal Theoretical Analysis Methods

The current computational methods for solving the energy band structure of PCs include the plane wave expansion method [18], the transfer matrix method [19], the multiple scattering theory method [20], the time domain difference method [8], and the finite element method [21]. Among them, the finite element method is widely used because of its clear concept, high applicability, good convergence, and its ability to accurately calculate the band gap properties of PCs. In this paper, the vibration isolation performance of PC plates is investigated based on COMSOL Multiphysics 5.6 finite element software.

2.2. Calculation Method of Band Structure

According to lattice theory and Bloch’s theorem [22], the energy band structure of an infinitely periodic PC in the ideal state can be obtained through setting Floquet period boundaries on the boundary of the finite element model of the PC, with stress-free boundary conditions invoked with the Z-direction free boundary surface. The fluctuation equation of anelastic wave propagating in a solid is Equation (1):

$${\displaystyle \sum _{j=1}^3\left\{\frac{\partial }{\partial x_i}(\lambda \frac{\partial u_j}{\partial x_j})+\frac{\partial }{\partial x_j}\left[\mu \left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)\right]\right\}}=\rho \frac{{\partial }^2{u}_i}{\partial t^2} (i,j=x,y,z)$$

(1)

where u is the position vector; ρ is material density; μ and λ are lamé constants; t is time.

According to the energy band theory, the ideal crystal has discrete translational periodicity and satisfies Bloch’s theorem for periodic systems, so the crystal structure can be simplified and only one single cell is required. The ideal crystal has point group symmetry, and the wave vector only needs to scan the whole integrable Brillouin zone boundary to obtain its energy band structure. That is, with different wave vectors scanning M→Γ→X→M, the displacement field can be expressed as Equation (2):

$$u\left(r,t\right)=u_k\left(r\right){\mathrm{e}}^{i\left(kr-{\omega }^{2}t\right)}$$

(2)

where t is time; r is the position vector of boundary node; k is Bloch wave vector; $\omega$ is the eigenvalue.

The idea of solving the energy band structure using the finite element method is to obtain the generalized eigenvalue of Equation (2) through solving the fluctuation equation (Equation (1)) and its energy band structure through solving Equation (3).

$$Ku=Mu{\omega }^2$$

(3)

The real stiffness matrix of the entire structure is denoted as the letter K, and the real mass matrix as the letter M. The entire structure’s eigenvector is designated as the letter u.

2.3. Phonon Crystal Structure Design

The PC proposed in this paper is a local-resonance-type PC. As shown in Figure 1, the PC consists of a steel cylinder, a silicone rubber cladding layer, a steel sleeve, a functional gradient material shell, and an epoxy resin connection plate. In other words, the steel cylinder core is covered with a silicone rubber cladding layer, the steel sleeve is connected with four epoxy resin connection plates, and finally, the functional gradient material shell is applied. The specific parameters of this PC are shown in

Table 1. Geometric parameters of the new phonon crystal.

a/mm	R1/mm	R2/mm	R3/mm	R4/mm	b/mm	c/mm	d/mm
35	8	9	11	13	2.2	1	70

and

Table 2. Material parameters of the new phonon crystal.

Material	Density (kg·m−3)	Young’s Modulus E/MPa	Poisson’s Ratio ν
Epoxy resin	1180	4350	0.368
Silicone rubber	1300	0.1175	0.469
Steel	7780	210,600	0.300
Aluminum	2730	77,600	0.352
Concrete	2300	25,000	0.200

. Steel and epoxy resin are chosen as the base material of the functional gradient material, and the finite element calculation of the physical parameters is carried out using the linear mixing law (Vought model) [23,24] as in Equation (4).

$$T=T_{\mathrm{c}}{R}_{\mathrm{c}}+Y_v{Q}_v$$

(4)

where T is the physical property of a position of functionally graded material; T_c is the physical property of material 1; Y_v is the physical property of material 2; R_c is the volume fraction of material 1; Q_v is the volume fraction of material 2.

The material parameter transition function adopted is shown in Equation (5) as a first-order function varying with the coordinate axis ($0\le x\le a$).

$$T=T_{\mathrm{c}}0.007\left(x\right)+Y_v\left[1-0.007\left(x\right)\right]$$

(5)

The square PC integrable Brillouin zone is shown in Figure 2. The phonon crystal grid division is shown in Figure 3.

3. Band Structure

3.1. Band Structure Analysis

The energy band structure of the PC was obtained through scanning the integrable Brillouin zone using COMSOL Multiphysics 5.6 software as shown in Figure 4.

Based on the band structure of the PC shown in Figure 4, it can be observed that the functional gradient material PC exhibits two distinct BGs within the frequency range of 0 to 150 Hz. The first BG starts at 0.00 Hz and ends at 65.81 Hz, with a width of 65.81 Hz. The second BG starts at 79.95 Hz and ends at 128.32 Hz, with a width of 48.37 Hz. The total BG width is 114.18 Hz, accounting for 76.12% within this frequency range. Existing studies found [25] that the structural vibration caused by a traffic load is between 0 and 100 Hz, so the functional gradient material PC designed in this paper can effectively suppress the vibration caused by a traffic load.

3.2. Vibration Mode Analysis

Three points A₁, A_2, and S₁ in Figure 4 were selected for vibration modal analysis, as shown in Figure 5, where A₁ and A₂ are located at the first BG and the second BG cutoff frequency, and S₁ is located at the start frequency of the second BG.

Near the BG cutoff frequency, a local resonant coupling is formed between the oscillator and the PC shell, causing them to vibrate in a collaborative manner. This resonant coupling causes the oscillator and the PC shell to move together at the BG cutoff frequency and form a stable vibration mode. However, at the initial frequency, due to frequency mismatch or weak coupling, the resonance coupling effect between the oscillator and the PC shell is weakened, resulting in the vibration of only the oscillator itself, without the common vibration mode.

In summary, the formation of a BG is due to resonance coupling between the oscillator and the PC shell caused by the local resonance effect near the BG cut-off frequency, while at the initial frequency, only the vibration of the oscillator itself is caused by the weakened coupling.

4. Analysis of Vibration Transmission Characteristics of Phonon Crystal Plate

4.1. Experimental Simulation of Vibration Transmission Response

In order to further verify the existence of an elastic wave gap in the finite PC structure, displacement excitation was applied to the left side of the PC plate in different directions (see Figure 6), and the vibration transmission response curves of bending wave and longitudinal wave were obtained, as shown in Figure 7. (The amplitude frequency response function is $TL=20\lg \left(R_{output}/R_{input}\right)$).

As shown in Figure 7, the bending wave transmission curve of the PC plate attenuates obviously in the range of 0~87.71 Hz and 100.01~197.5 Hz, and the maximum attenuation amplitude is −20.86 db. This indicates that the bending vibration is effectively controlled in this region. At the same time, it can be observed from the longitudinal transmission curve that obvious attenuation exists in the range of 0~99.5 Hz and 99.8~195.5 Hz. This shows that longitudinal vibration is also effectively controlled in this region. The BG range of the band structure is completely contained in the attenuation domain of the vibration transfer curve of the curved wave and the longitudinal wave. The complete BG of the PC is generated by the coupling of the curved BG and the longitudinal band gap.

4.2. Vibration Mode Analysis

In order to further explain the cause of BG formation, three points of the curved wave transmission curve ${\mathrm{A}}_1^{-}$, ${\mathrm{A}}_2^{-}$, ${\mathrm{S}}_1^{-}$ and three points of the longitudinal wave transmission curve ${\mathrm{A}}_1^{*}$, ${\mathrm{A}}_2^{*}$, ${\mathrm{S}}_1^{*}$ are selected for vibration modal analysis, as shown in Figure 8.

The points ${\mathrm{A}}_1^{-}$ and ${\mathrm{A}}_1^{*}$ are located at the cutoff frequency of the first attenuation domain, at the interaction between the oscillator and the PC plate, where the vibration of the oscillator excites the local resonance response in the lattice. When a vibrator vibrates downward, the lattice adjacent to it is excited and deflected downward, causing local vibration. Conversely, when the vibrator vibrates upward, the adjacent lattice will produce opposite displacement. This local resonance effect leads to the interaction between the vibrator and the lattice and forms a phase difference between the vibrator and the lattice in the region of periodic distribution. The vibrator at point ${\mathrm{A}}_1^{*}$ swings upward to the left, unable to form a resultant force with the elastic wave propagating in the PC plate. At this time, the vibration of the vibrator is in resonance with the PC plate, and the vibration is gradually strengthened.

The points ${\mathrm{S}}_1^{-}$ and ${\mathrm{S}}_1^{*}$ are located at the initial frequency of the second attenuation domain. It can be seen from the vibration mode diagram that the vibration modes of the two are very similar. Both vibrators vibrate up and down, and the vibration of the vibrator stimulates the local resonance response in the lattice. The local resonance effect dominates, forming a coupling effect with the elastic wave propagating in the PC plate, at which time the vibration gradually weakens.

The points ${\mathrm{A}}_2^{-}$ and ${\mathrm{A}}_2^{*}$ are located at the cutoff frequency of the second attenuation domain. It can be seen from the vibration mode diagram of the two points that part of the vibrators in the PC plate vibrate irregularly, the local resonance effect is weakened, and they cannot be coupled with the elastic wave. At this time, the vibration begins.

5. Stress Analysis

A PC plate model was established in COMSOL, as shown in Figure 9. A force excitation perpendicular to the phonon crystal model was applied to the left side of the PC model, and the stress distribution of the phonon crystal model was analyzed via sweeping frequency on the right side.

Figure 10 shows the stress variation curves of the phonon crystal model. The stress modes of two points K and L are selected for analysis, where point K is located in the BG and point L is located in the passband, as shown in Figure 11.

According to the analysis of the stress distribution diagram in Figure 11, point K is within the BG range. It can be observed from the figure that the stress is mainly concentrated in the first three cycles and does not propagate to the rear. In addition, compared with point L, the maximum stress at point K is lower, only one percent of that at point L. It is worth noting that the stress distribution at point K is relatively uniform and there is no stress concentration phenomenon.

In contrast, point L lies within the passband, and a red circle can be seen in the figure indicating a large concentration of stress around this point. It can be observed from the figure that there is a large stress concentration in a few vibrator regions near point L.

In general, the stress distribution in the forbidden zone is relatively wide and uniform, and there is no stress concentration phenomenon. On the contrary, in the passband, the stress distribution is not uniform and a large concentration of stress occurs. Therefore, the stress can be adjusted through adjusting the BG.

6. Application Simulation of Phonon Crystals with Functionally Graded Materials

6.1. Structural Design

The PC plate made in this paper can be used in the bottom plate of a bridge box girder, and the specific application position is shown in Figure 12. In order to simulate the practical application, a miniature box girder model is established using COMSOL 5.6 software. Vibration transmission excitation results are shown in Figure 13, and size parameters are shown in

Table 3. Structural dimensions.

h1/mm	h2/mm	h3/mm	h4/mm	h5/mm	e/mm	g/mm	f/mm	L/mm
320	60	50	70	10	125	10	30	210

.

6.2. Vibration Transmission Simulation Analysis

As can be seen from the vibration transfer curve of the box girder in Figure 13, both a curved wave and longitudinal wave exist in the attenuation domain, which is consistent with the vibration transfer curve of the phonon crystal plate in Figure 7, indicating that the box girder effectively inhibits the propagation of vibration in the forbidden band. In order to further analyze the vibration mode of the structure, the vibration modes of point O on the curved wave transmission curve and point P on the longitudinal wave transmission curve were selected for analysis (as shown in Figure 14). Both points are in the first attenuation domain, where point O is controlled via bending vibration and point P via longitudinal vibration. From the two-point vibration modes, it can be seen that the motion direction of the oscillators in the two PC plates is opposite to the direction of vibration excitation, and the coupling force is formed with the elastic wave propagating in the box girder, thus canceling the vibration. The vibration damping performance of the phonon crystal plate can be effectively verified through the application simulation of the miniature box girder.

7. Conclusions

In this paper, a functionally graded material phonon crystal plate is designed, and the vibration damping performance of the functionally graded material phonon crystal plate is simulated using a finite element numerical simulation method. The results are as follows:

The band structure of phonon crystal plates with functionally gradient materials has two complete band gaps within 0~150 Hz, among which the initial frequency of the first band gap is 0.00 Hz, the total band gap cut-off frequency is 128.32 Hz, and the total band gap width is 114.18 Hz, accounting for 76.12% in the band gap range. The band gap width between 0 and 100 Hz is 94.13 Hz, which can effectively suppress the vibrations caused by traffic loads.

The vibrational transmission characteristics of this new PC single cell with 6 × 3 periods in X and Y directions are analyzed, and the results completely verify the existence of the forbidden band, prove the effective suppression of elastic waves from 0 to 150 Hz in this new PC, and point out the reasons for the generation of the forbidden band in the PC.

Stress analysis of the functionally graded material phonon crystal plate shows that functionally graded material can reduce the stress concentration of a PC plate. Finally, a miniature box girder model of the PC plate with functionally graded materials was established and the vibration transmission response was analyzed. The results show that the PC plate still has a good damping effect. The research results provide a new design method for vibration and noise reduction in traffic engineering.