Band-gap characteristics of elastic metamaterial plate with axial rod core by the finite element and spectral element hybrid method

A Finite/Spectral Element Hybrid Method for Modeling and Band-Gap Characterization of Metamaterial Sandwich Plates

“…The spectral stiffness matrix of the tensional element (rod element) of the axially deformed beam can be derived when only the axial force and the axial displacement are considered for the beam. The relationship between the nodal axial forces F r 1 and F r 2 , and the nodal axial displacements W r 1 and W r 2 in the frequency domain can be written as [ 41 , 42 , 43 ] where is the 2 × 2 spectral stiffness matrix of the tensional element, which has the following form: where is the longitudinal wave number.…”

“…For the two thin cover plates, considering only the out-of-plane deformation, the dynamic stiffness matrix of a Kirchhoff plate element is a 12 × 12 matrix, and it can be derived as [ 41 , 42 ] where is the stiffness matrix, is the mass matrix, is the volume of the plate element, B pb is the second-order partial derivative matrix of the shape functions, R pb is the flexural rigidity matrix and N p is the shape function matrix.…”

“…Each one of the FEM and SEM has its own limitation, but when they are combined to create the finite/spectral element hybrid method (FE-SEHM), they can make full use of their advantages. As the one-dimensional rods act as the local resonators and the irregular or complex matrices are analyzed by the SEM and FEM, respectively, the FE-SEHM can achieve unrestricted structural dimensions, convenient modeling, few element numbers and improved calculation accuracy and efficiency [ 41 , 42 ].…”

“…In this paper, on the basis of [ 41 ], wherein the core rods are assumed to have only axial deformations and only the transverse deformations of the cover plates are considered for the elastic metamaterial sandwich plates with rod cores, the FE-SEHM is further extended to investigate the band-gap characteristics of locally resonant elastic metamaterial sandwich plates with axially loaded Timoshenko beam cores considering the out-of-plane and in-plane vibrations of the thin face plates. The dynamic stiffness matrices of the beam elements and the plate elements are derived by the SEM and the FEM, respectively, and according to the nodal coordinate relationships, they are assembled to the whole dynamic stiffness matrix of the metamaterial sandwich plate.…”

“…The FRFs of the elastic metamaterial sandwich plate are calculated. The axial vibration band-gaps of the beams are calculated and compared with those of the metamaterial plates with rod cores [ 41 ], and the transverse vibration band-gaps of the beams are also obtained and analyzed.…”