Applicability Two-Dimensional Differential Integral Quadrature Method in Vibration Analysis of Multi-Directional Functionally Graded Porous Viscoelastic Plates

Abstract

This study formulates a differential integral quadrature method (DIQM) to analyze the free vibration characteristics of multi-directional functionally graded material (MFGM) viscoelastic porous plates. The kinematic relations are derived using a unified shear deformation plate theory, while material behavior is governed by the integer-order Kelvin-Voigt viscoelastic constitutive model. Power-law functions define the spatial gradation of material constituents along the length, width, and thickness directions. Two distinct porosity distributions are incorporated to characterize void and cavity variations through the plate's thickness. Hamilton's variational principle yields five coupled governing equations expressed as partial differential equations with variable coefficients. The differential quadrature method (DQM) discretizes these governing equations, with integral quadrature method (IQM) efficiently resolving the variable coefficients. This discretization results in an algebraic system constituting a quadratic eigenvalue problem. The eigenvalues' real and imaginary components provide the damping coefficients and natural frequencies, respectively. The proposed model and solution methodology are validated against established unified shear formulations, MFGM porous plates, and viscoelastic plate solutions available in literature. Comprehensive parametric studies systematically investigate the influence of material gradation indices, porosity parameters, boundary conditions, and viscoelastic coefficients on the natural vibration response of thick MFGM viscoelastic porous plates. The results demonstrate that an increase in either of the material gradation indices leads to a decrease in both of the real and imaginary parts of the fundamental frequencies. © 2025 John Wiley & Sons Ltd.