Vibration of an orthotropic nanoplate resting on a viscoelastic foundation: nonlocal and fractional derivative viscoelasticity approach

Abstract

Here, we investigate the free vibration behavior of a nanoplate resting on a foundation with viscoelastic properties using nonlocal elasticity and fractional viscoelasticity approach. Nanoplate is modeled using nonlocal and fractional viscoelastic constitutive equation and orthotropic Kirchhoff-Love plate theory. Viscoelastic foundation is represented by the viscoelastic model with fractional derivative operator. Governing equation is derived using D’Alambert’s principle and solution is assumed in terms of Fourier series using separation of variables method and satisfying the simply supported boundary conditions for nanoplate. Fractional differential equation is solved using the Laplace and Mellin-Fourier transforms and residue theory. Complex poles of unknown function are determined by finding the roots of the characteristic equation using technique that is available in the literature. In order to show the effect of fractional derivative parameters, damping coefficients and nonlocal parameter on complex roots i.e. damped frequency and damping ratio as well as on nanoplate’s displacement, few numerical examples are given.