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Stochastic Analysis of a Nonlocal Fractional Viscoelastic Bar Forced by Gaussian White Noise

[+] Author and Article Information
G. Alotta

Engineering and Architecture Faculty,
University of Enna “Kore,”
Viale delle Olimpiadi,
Enna 94100, Italy
e-mail: gioacchino.alotta@unikore.it

G. Failla

Departiment of Civil, Energy, and Environment,
Materials Engineering (DICEAM),
University of Reggio Calabria,
Via Graziella, Località Feo di Vito,
Reggio Calabria 89124, Italy
e-mail: giuseppe.failla@unirc.it

F. P. Pinnola

Department of Innovation Engineering,
Università del Salento,
Edificio La Stecca, S.P. 6 Lecce-Monteroni,
Lecce 73100, Italy
e-mail: francesco.pinnola@unisalento.it

Manuscript received June 22, 2016; final manuscript received October 16, 2016; published online June 12, 2017. Assoc. Editor: Ioannis Kougioumtzoglou.

ASME J. Risk Uncertainty Part B 3(3), 030904 (Jun 12, 2017) (7 pages) Paper No: RISK-16-1094; doi: 10.1115/1.4036702 History: Received June 22, 2016; Revised October 16, 2016

Recently, a displacement-based nonlocal bar model has been developed. The model is based on the assumption that nonlocal forces can be modeled as viscoelastic (VE) long-range interactions mutually exerted by nonadjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo's fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the nonlocal fractional viscoelastic bar introduced in previous papers, discretized with the finite element method (FEM), forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.

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Figures

Grahic Jump Location
Fig. 2

Pure axial deformation η and long range forces

Grahic Jump Location
Fig. 3

Axial equilibrium of a bar segment; positive sign conventions are reported

Grahic Jump Location
Fig. 4

Displacement variance at x = L (black) and x = L/2 (gray) for elastoviscous (EV) case α = 1/4

Grahic Jump Location
Fig. 5

Displacement variance at x = L (black) and xL/2 (gray) for VE case α = 3/4

Grahic Jump Location
Fig. 6

PSD of axial displacement at x = L (black) and x = L/2 (gray) for EV case α = 1/4

Grahic Jump Location
Fig. 7

PSD of axial displacement at x = L (black) and x = L/2 (gray) for VE case α = 3/4

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