Dynamics of Nonlocal Rod by Means of Fractional Laplacian

Abstract

The use of fractional models to analyse nonlocal behaviour of solids has acquired great importance in recent years. The aim of this paper is to propose a model that uses the fractional Laplacian in order to obtain the equation ruling the dynamics of nonlocal rods. The solution is found by means of numerical techniques with a discretisation in the space domain. At first, the proposed model is compared to a model that uses Eringen’s classical approach to derive the differential equation ruling the problem, showing how the parameters used in the proposed fractional model can be estimated. Moreover, the physical meaning of the model parameters is assessed. The model is then extended in dynamics by means of a discretisation in the time domain using Newmark’s method, and the responses to different dynamic conditions, such as an external load varying with time and free vibrations due to an initial deformation, are estimated, showing the difference of behaviour between the local response and the nonlocal response. The obtained results show that the proposed model can be used efficiently to estimate the response of the nonlocal rod both to static and dynamic loads.

Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements

“…The output process is then obtained by applying the Duhamel superposition integral, and finally the displacement samples are processed to evaluate the non-stationary variance. Further contributions exploring dynamics of nonlocal continua are provided in [108], where nanobeam-based resonators are investigated, in [109], concerning forced vibrations of dielectric elastomer-based microcantilevers, in [110], where buckling of graphene platelet-reinforced nanostructures is examined, in [111], which concerns dynam-ics of a piezoelectric semiconductor nanoplate, in [112], in which dynamics of nonlocal rods is examined, in [113], where transverse vibrations of nanobeams with multiple cracks are evaluated, and in [114], in which mechanical static and dynamic behaviors of microsystems are investigated providing fundamental concepts of modeling and design.…”