Limit behaviour of Eringen's two-phase elastic beams

In this paper, the bending behaviour of small-scale Bernoulli-Euler beams is investigated by Eringen's two-phase local/nonlocal theory of elasticity. Bending moments are expressed in terms of elastic curvatures by a convex combination of local and nonlocal contributions, that is a combination w...

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Bibliographic Details
Published in:arXiv.org 2021-07
Main Authors: Vaccaro, Marzia Sara, Pinnola, Francesco Paolo, Francesco Marotti de Sciarra, Barretta, Raffaele
Format: Article
Language:English
Subjects:
Bending moments
Boundary conditions
Convolution
Convolution integrals
Elastic analysis
Elastic beams
Elastic limit
Euler-Bernoulli beams
Mathematical analysis
Nonlocal elasticity
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Summary:In this paper, the bending behaviour of small-scale Bernoulli-Euler beams is investigated by Eringen's two-phase local/nonlocal theory of elasticity. Bending moments are expressed in terms of elastic curvatures by a convex combination of local and nonlocal contributions, that is a combination with non-negative scalar coefficients summing to unity. The nonlocal contribution is the convolution integral of the elastic curvature field with a suitable averaging kernel characterized by a scale parameter. The relevant structural problem, well-posed for non-vanishing local phases, is preliminarily formulated and exact elastic solutions of some simple beam problems are recalled. Limit behaviours of the obtained elastic solutions, analytically evaluated, studied and diagrammed, do not fulfill equilibrium requirements and kinematic boundary conditions. Accordingly, unlike alleged claims in literature, such asymptotic fields cannot be assumed as solutions of the purely nonlocal theory of beam elasticity. This conclusion agrees with the known result which the elastic equilibrium problem of beams of engineering interest formulated by Eringen's purely nonlocal theory admits no solution.
ISSN:2331-8422
DOI:10.48550/arxiv.2107.12731