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

[+] Author and Article Information
Giuseppina Autuori

Department of Mathematics and Informatics,
University of Perugia,
Via Vanvitelli 1,
Perugia 06123, Italy
e-mail: giuseppina.autuori@unipg.it

Federico Cluni

Department of Civil and
Environmental Engineering,
University of Perugia,
Via Duranti 93,
Perugia 06125, Italy
e-mail: federico.cluni@unipg.it

Vittorio Gusella

Department of Civil and
Environmental Engineering,
University of Perugia,
Via Duranti 93,
Perugia 06125, Italy
e-mail: vittorio.gusella@unipg.it

Patrizia Pucci

Department of Mathematics and Informatics,
University of Perugia,
Via Vanvitelli 1,
Perugia 06123, Italy
e-mail: patrizia.pucci@unipg.itt

Manuscript received June 21, 2016; final manuscript received May 10, 2017; published online June 12, 2017. Assoc. Editor: Mario Di Paola.

ASME J. Risk Uncertainty Part B 3(3), 030902 (Jun 12, 2017) (5 pages) Paper No: RISK-16-1092; doi: 10.1115/1.4036806 History: Received June 21, 2016; Revised May 10, 2017

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.

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Fig. 2

Influence of s on displacements u

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Fig. 1

Rod loaded with a distributed continuous force f

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Fig. 3

Histogram of the generated values of s

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Fig. 4

Histogram of estimated values of u at midspan

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Fig. 5

Cumulative distribution function and PDF for displacements at x = 0

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Fig. 6

Cumulative distribution function and PDF for displacements at x=L/2




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