Mikhasev, G, Avdeichik, E and Prikazchikov, D (2018) Free vibrations of nonlocally elastic rods. Mathematics and Mechanics of Solids. ISSN 1081-2865

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Abstract

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.

Item Type: Article
Additional Information: This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Sage at https://doi.org/10.1177/1081286518785942. Please refer to any applicable terms of use of the publisher.
Uncontrolled Keywords: Eringen’s nonlocal elasticity, two-phase integral model, nanorod, free longitudinal vibrations, asymptotic method, natural frequencies
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Natural Sciences > School of Computing and Maths
Depositing User: Symplectic
Date Deposited: 08 Oct 2018 08:44
Last Modified: 08 Oct 2018 08:44
URI: http://eprints.keele.ac.uk/id/eprint/5407

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