Frequency dependent effective modulus of square grid lattice using spectral element method

A spectral element method-based formulation is developed to obtain the frequency-dependent effective Young’s and shear modulus of a two-dimensional square grid lattice. The methodology presented in this work is a simplified approach to obtain effective parameters. The lattice is assumed to be constituted of Euler–Bernoulli beam elements which are arranged in a specific order to obtain the said geometry. Spectral element formulation of the beam element considering axial and flexural deformation is presented, which in due course is employed to obtain the global spectral element matrix of the unit cell. A detailed procedure to obtain the frequency-dependent stiffness matrix of the unit cell is presented on which proper boundary conditions are applied using specific transformation matrices for each case. The frequency-dependent properties are also calculated using the software package COMSOL Multiphysics to validate the results. Frequency responses of the unit cell are obtained using the proposed methodology as well as the software package, and the results are compared. It is observed that the proposed methodology predicts the peaks at nearly the same frequency range as obtained through the finite element solution obtained in COMSOL Multiphysics. Finally, the results are presented which show the frequency-dependent effective parameters for the unit cell. The methodology presented here opens up a new way to calculate the effective properties of two-dimensional lattice structures and with some modifications can also be extended to three-dimensional lattice structures.