Effects of the Fractional Laplacian Order on the Nonlocal Elastic Rod Response

Abstract

In this paper, we yield with a nonlocal elastic rod problem, widely studied in the last decades. The main purpose of the paper is to investigate the effects of the statistic variability of the fractional operator order s on the displacements u of the rod. The rod is supposed to be subjected to external distributed forces, and the displacement field u is obtained by means of numerical procedure. The attention is particularly focused on the parameter s, which influences the response in a nonlinear fashion. The effects of the uncertainty of s on the response at different locations of the rod are investigated by the Monte Carlo simulations. The results obtained highlight the importance of s in the probabilistic feature of the response. In particular, it is found that for a small coefficient of variation of s, the probability density function of the response has a unique well-identifiable mode. On the other hand, for a high coefficient of variation of s, the probability density function of the response decreases monotonically. Finally, the coefficient of variation and, to a small extent, the mean of the response tend to increase as the coefficient of variation of s increases.