Mathematical modeling of free vibration of star-shaped auxetic rectangular plate

Abstract

This paper dealt with the vibration characteristics of an auxetic rectangular plate under in-plane compression. Firstly, the equivalent bending stiffness matrix of a star-shaped auxetic plate was obtained using Castigliano's theorem and the homogenization technique. Then, employing the classical plate theory (CPT) in conjunction with the Rayleigh–Ritz method, the natural frequencies of auxetic plate were extracted. The chebyshev polynomial series has been selected to define the assumed displacement fields of the plate. Convergence study for the Rayleigh–Ritz method was conducted. The accuracy of the proposed mathematical model for the star-shaped auxetic plate was validated using the results from a finite element analysis (FEA). Effects of unit cell geometric parameters on the natural frequencies of the plate were examined. The auxeticity angle of star-shaped pattern had a significant effect on the natural frequencies. The present approach can be extended into other auxetic patterns, such as re-entrant bowtie auxetics.