Sensitivity determination of band gap of locally resonant phononic crystals based on joint grey correlation analysis

Abstract

The unique low-frequency band gap characteristics of locally resonant phononic crystals (LRPCs) make them have broad application value and prospect in the design of metamaterials for vibration and noise reduction. If the component material parameters and geometric parameters of the LRPCs are reasonably selected, it can open up high-quality low-frequency wide band gap, that is, the opening of the low-frequency locally resonant band gap is influenced by its component material parameters and geometric parameters. However, at present, most of the studies on the factors affecting the band gap of LRPCs are carried out from the qualitative point of view, and there is a lack of quantitative studies on the factors affecting the band gap of LRPCs, which limits the application of LRPCs in engineering, especially in the design of vibration and noise reduction metamaterials. In view of this, this paper quantitatively studies the factors affecting the band gap of LRPCs based on the joint grey correlation analysis method, that is, the sensitivity determination of the local resonance band gap. Firstly, the improved plane-wave expansion method is derived to calculate the band structure of LRPCs. Secondly, based on the theory of grey correlation analysis, the correlation degree of the factors affecting the band gap of the LRPCs are calculated. Finally, the weights of the factors affecting the band gap of the LRPCs are determined by the 9-scale analytic hierarchy process. The results show that the elastic modulus of the wrapper layer has the most significant impact on the local resonance band gap starting frequency, band gap cutoff frequency, and band gap width, followed by the scatterer radius, wrapper layer radius, scatterer density, and Poisson's ratio of the wrapper layer. Among them, the lattice constant, scatterer elastic modulus, scatterer Poisson's ratio, wrapper layer density, matrix density, matrix elastic modulus, and matrix Poisson's ratio have almost no effect on the starting frequency of the local resonance band gap; The elastic modulus of scatterer, Poisson's ratio of scatterer, wrapper layer density, matrix elastic modulus, and matrix Poisson's ratio have almost no effect on the local resonance band gap cutoff frequency and the local resonance band gap width. This paper quantitatively compares the influence of geometric and material parameters of LRPCs on the local resonance band gap, and quantifies the correlation between different influencing factors. The relevant research results of this paper can provide specific references and inspirations for the design and optimization of LRPCs metamaterials, improve their effectiveness in engineering applications, and make the optimization design more efficient.