Flexural wave propagation in periodic Micropolar-Cosserat panels: Spectral Element Formulation

Abstract

In this research, a perspective of the dispersion relation of flexural waves on periodic structures is proposed. The rotational theory of micro-elements is stem to explore the favorable impact of size effect on structures through the localization of the strain energy. The panels are modelled as a one-dimensional (1-D) Micropolar–Cosserat (MC) continuum to explore the effect of length-scale or micro-rotational waves of solids. The spectral element formulation of a panel and the transfer matrix of the unit cell is presented within the framework of the state-space method. The propagation constant in the eigenvalue domain is established based on the Bloch–Floquet theorem to account for the periodicity of the panel. The MC theory in analyzing the band-gap (stop-band) and pass-band characteristics of the unit cell panels is validated by a commercial finite element software package-COMSOL Multiphysics. The mode shape of unit cell for the start and end point of band-gap is further analyzed within the realms of finite element method (FEM) analysis. The appropriate limiting condition reverts MC spectral element into Timoshenko (TM) beam model. The comparison of Timoshenko beam model with the spectral element of 1-D plane-stress (PS) analysis is also investigated. It is observed that the shear wave of the MC model is coupled with a micro-rotational wave of micro-elements, unlike the classical continuum framework. The response of the finite structure subjected to various frequencies is presented too. Presence of vibration attenuation in the specified band-gap (BG) is justified through the elemental dynamic stiffness (DS) matrix of unit cells. This reduced the level of vibration at the specific frequency range within the attenuation band or disintegrates it into two bands. The attenuation bandwidth and its location is significantly affected by the modulation of both geometry and material properties of the beam. Investigation of flexural wave propagation based on the proposed 1-D MC beam theory shows good agreement with the two-dimensional (2-D) plane-stress FEM analysis.