Viscoelastic free vibration analysis of in-plane functionally graded orthotropic plates integrated with piezoelectric sensors: Time-dependent 3D analytical solutions

Abstract

This paper proposes an accurate three-dimensional framework for elastic and viscoelastic free vibration investigation of in-plane functionally graded (IPFG) orthotropic rectangular plates integrated with piezoelectric sensory layers. The developed analytical framework is capable of considering layer-wise unidirectional linear functional gradation in both stiffness and density of the orthotropic composite layers. 3D piezoelasticity-based governing equations of motion are formulated in mixed form by employing Hamilton’s principle, and further solved analytically for Levy-type support conditions using the power-series-based extended Kantorovich method (EKM) jointly with Fourier series. The displacements, stresses, and electrical variables (electric field and electric potential) are solved as the primary variables that ensure the point-wise interlayer continuity and electro-mechanical support conditions. The viscoelastic property of the orthotropic interlayer is defined by employing Biot model, which is similar to the standard linear viscoelastic model. The correctness and efficacy of the present mathematical model are established by comparing the present numerical results with published literature and 3D finite element results, obtained by utilizing user material subroutine in the commercial FE software ABAQUS. An extensive numerical study is performed for various configurations and thickness ratios to investigate the influences of in-plane gradation, viscoelasticity and their coupled effects on the free-vibration response of hybrid laminated plates. It is found that in-plane gradation of stiffness and density remarkably alters the flexural frequencies and corresponding mode shapes of the hybrid intelligent rectangular plates. The flexural frequencies and stresses in the plate can be modified by selecting suitable grading indexes. Another interesting observation is that the in-plane gradation shows a considerably less effect on the electrical response of piezoelectric layers, which can play a vital role in the design of sensors and actuators for dynamic applications. Further, the numerical study demonstrates a potential time-dependent structural behaviour based on the present viscoelastic modelling. The consideration of viscoelasticity could be crucial for analysing the mechanical behaviour of a wide range of polymer composites more realistically and for prospective temporal programming in smart structural systems by exploiting the viscoelastic effect. Although the present analytical solution has been proposed for the free-vibration investigation of smart in-plane functionally graded (IPFG) viscoelastic plates, it can also be utilized directly to analyze the symmetric and asymmetric laminated piezoelectric smart plates with constant properties.