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.