# Design of a Functionally Graded Material Phonon Crystal Plate and Its Application in a Bridge
## 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].  
[PERSON] 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.  
[PERSON] 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.  
[PERSON] 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.