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.
Highlights
Classical elasticity theory assumes that the stress at a point depends only on the displacements in the neighbourhood of the point itself
When dealing with elements at the micro-scale, and in order to avoid formulating the problem in the context of lattice theory, several authors prefer to work within the framework of continuum mechanics by introducing the theory of nonlocal elasticity, the roots of which can be traced to the work of Eringen and Edelen [1]
An approach to analyse the dynamics of a rod with a nonlocal elastic behaviour is proposed
Summary
Classical elasticity theory assumes that the stress at a point depends only on the displacements in the neighbourhood of the point itself. The results obtained are coherent and meaningful when dealing with structural elements at the macro-scale, i.e., in which the dimensions are much larger than the scale of elementary material particles. At the micro-scale, the classical theory of elasticity may be not adequate to estimate the response of solids. Bazant [2] showed how the damage may be considered a nonlocal phenomenon, while in [3] it is shown that the effect of the micro-structure can be relevant in the case of wave dispersion in one-dimensional solid. In [4] it is shown that nonlocality can arise at the meso-scale when dealing with composite material
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