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Free vibrations of nonlocally elastic rods

Mikhasev, G.; Avdeichik, E.; Prikazchikov, D.

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G. Mikhasev

E. Avdeichik


Several of the Eringen’s nonlocal stress models, including two-phase and purely nonlocal integral
models, along with the simplified differential model, are studied in case of free longitudinal vibrations
of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel
in the nonlocal integral models, the integro-differential equation corresponding to the two-phase
nonlocal model is reduced to a fourth order differential equation with additional boundary conditions
taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and
asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies
and associated modes found in the framework of the purely nonlocal model and its ”equivalent”
differential analogue are also compared. A detailed analysis of solutions suggests that the purely
nonlocal and differential models lead to ill-posed problems.

Journal Article Type Article
Acceptance Date Jun 7, 2018
Online Publication Date Jul 13, 2018
Publication Date May 1, 2019
Publicly Available Date May 26, 2023
Journal Mathematics and Mechanics of Solids
Print ISSN 1081-2865
Publisher SAGE Publications
Peer Reviewed Peer Reviewed
Volume 24
Issue 5
Pages 1279-1293
Keywords Eringen’s nonlocal elasticity, two-phase integral model, nanorod, free longitudinal vibrations,asymptotic method, natural frequencies
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