Nonlocal higher-order finite element modeling for vibration analysis of viscoelastic orthotropic nanoplates resting on variable viscoelastic foundation

Abstract

This article develops a novel finite element formulation based on nonlocal theory to analyze the vibration of viscoelastic orthotropic nanoplates resting on the variable viscoelastic foundation (VEF). The mechanical properties of nanoplate are assumed to be viscoelastic orthotropic according to Kelvin’s model. The variable VEF consists of two layers: a shear layer with constant stiffness, the other layer is described as a system composed of alternating damping and springs, in which the spring stiffness and the damping coefficient vary x-axis only. Based on Hamilton’s principle, a refined higher-order shear deformation plate theory (HSDT) and nonlocal theory are used to establish motion equations of the nanoplates. A four-node quadrilateral finite element is created by combining two interpolation functions including Lagrangian and Hermitian interpolation functions to describe the primary variables corresponding to the in-plane displacements and transverse displacement. The obtained results in our work are verified through reliable publications. A series of factors influencing the vibration of orthotropic nanoplates resting on the variable VEF is discovered and discussed.