Free vibrations of higher-order quasi-3D viscoelastic bi-directional functionally graded plates

Abstract

This study introduces a quasi-3D shear deformation theory to analyse the coupled eight-parameter dynamics of bi-directional functionally graded Kelvin-Voigt viscoelastic plates. By focusing on the quasi-3D formulation, this work uniquely captures the influence of continuous axial and thickness gradation in material properties, utilising a power-law rule to determine effective properties. Viscoelasticity is modelled using the Kelvin-Voigt model to incorporate the energy dissipation of the composite plate structure. Eight governing equations, which are coupled through the in-plane and out-of-plane motions including stretching, are derived via Hamilton’s variational principle. A multi-modal discretisation is conducted using a weighted-residual method as the proposed solution for the in-plane and out-of-plane displacements. Thereafter, a numerical technique is employed to solve the resultant equations, obtaining both the real and imaginary parts of the natural frequencies. The proposed model is validated for the natural frequencies with an elastic counterpart from the literature; a homogenous version of the model is also validated against literature. The results obtained not only provide a comprehensive understanding of how the bi-directional functionally graded viscoelastic material and geometrical parameters influence the coupled eight-parameter dynamics of composite straight plates, but also propose a reliable benchmark for such systems through a quasi-3D model. The investigation revealed that the difference in frequency resulting from the application of both 2D and quasi-3D theories is most dominant in the case of thick plates.