Authors: Challamel, Noël
Zorica, Dušan 
Atanacković, Teodor
Spasić, Dragan
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On the fractional generalization of Eringen's nonlocal elasticity for wave propagation
Journal: Comptes Rendus - Mecanique
Volume: 341
Issue: 3
First page: 298
Last page: 303
Issue Date: 1-Mar-2013
Rank: M22
ISSN: 1631-0721
DOI: 10.1016/j.crme.2012.11.013
Abstract: 
A fractional nonlocal elasticity model is presented in this Note. This model can be understood as a possible generalization of Eringen's nonlocal elastic model, with a free non-integer derivative in the stress-strain fractional order differential equation. This model only contains a single length scale and the fractional derivative order as parameters. The kernel of this integral-based nonlocal model is explicitly given for various fractional derivative orders. The dynamical properties of this new model are investigated for a one-dimensional problem. It is possible to obtain an analytical dispersive equation for the axial wave problem, which is parameterized by the fractional derivative order. The fractional derivative order of this generalized fractional Eringen's law is then calibrated with the dispersive wave properties of the Born-Kármán model of lattice dynamics and appears to be greater than the one of the usual Eringen's model. An excellent matching of the dispersive curve of the Born-Kármán model of lattice dynamics is obtained with such generalized integral-based nonlocal model.
Keywords: Born-Kármán model | Dispersive properties | Eringen model | Fractional derivative | Heterogeneous material | Nanostructures | Nonlocal elasticity | Scale effects | Wave propagation | Waves
Publisher: Elsevier
Project: Viscoelasticity of fractional type and shape optimization in a theory of rods 
Mechanics of nonlinear and dissipative systems - contemporary models, analysis and applications 
Secretariat for Science of Vojvodina, Grant 114-451-2167
European Community’s Seventh Framework Programme (FP7/2007–2013), Grant No. PIEF-GA-2010-271610

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