G. Mikhasev
Free vibrations of nonlocally elastic rods
Mikhasev, G.; Avdeichik, E.; Prikazchikov, D.
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.
| 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 |
| DOI | https://doi.org/10.1177/1081286518785942 |
| Keywords | Eringen’s nonlocal elasticity, two-phase integral model, nanorod, free longitudinal vibrations,asymptotic method, natural frequencies |
| Publisher URL | https://doi.org/10.1177/1081286518785942 |
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https://creativecommons.org/licenses/by-nc/4.0/
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