Nonlocal effects on the dynamic analysis of a viscoelastic nanobeam using a fractional Zener model

Abstract

• Improved nonlocal fractional Zener model, based on a hereditary integral constitutive law is proposed. • Motion equation of a nonlocal fractional nanobeam subjected to a uniform distributed load is determined. • Solution is obtained by using Galerkin's method, Laplace transforms, Bessel functions and binomial series . • Validity of the model is done by reducing the nonlocal study to the classic one and the analysis of the free vibrations . • Many graphs highlight the influence of nonlocal theory in the design of nanostructures. Using Eringen's nonlocal theory, a fractional dynamic analysis of a simply supported viscoelastic nanobeam is presented. The existence of a significant internal damping for the viscoelastic nanostructures led to the choice of a Zener model to obtain the governing equation. The solving of this is made with the help of an algorithm based on the Laplace transform, Bessel functions theory and the binominal series. The graphical representations show how the existence of the fractional derivative in the selected rheological influence the local and nonlocal dynamic response of a single-walled carbon nanotube (SWCNT). The validation study was performed by comparing the numerical results with the corresponding ones existing in the literature.