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A Study of Free High-Frequency Vibrations of an Inhomogeneous Nanorod, Based on the Nonlocal Theory of Elasticity

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Abstract

Free high-frequency longitudinal vibrations of an inhomogeneous nanosize rod are studied on the basis of the nonlocal theory of elasticity. The upper part of the spectrum with a wavelength comparable to the internal characteristic dimension of the nanorod is investigated. An integral-form equation with a Helmholtz kernel, containing both local and nonlocal phases, is used as the constitutive one. The original integrodifferential equation is reduced to a fourth-order differential equation with variable coefficients and a pair of additional boundary-value conditions is obtained. Using the WKB-method, we construct a solution of the boundary-value problem in the form of the superposition of a main solution and edge-effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model, providing an estimate of the upper part of the spectrum of eigenfrequencies.

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Authors and Affiliations

  1. Belarusian State University, 220030, Minsk, Belarus

    G. I. Mikhasev

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  1. G. I. Mikhasev
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Correspondence to G. I. Mikhasev.

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Translated by A. Muravnik

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Mikhasev, G.I. A Study of Free High-Frequency Vibrations of an Inhomogeneous Nanorod, Based on the Nonlocal Theory of Elasticity. Vestnik St.Petersb. Univ.Math. 54, 125–134 (2021). https://doi.org/10.1134/S1063454121020060

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  • DOI: https://doi.org/10.1134/S1063454121020060

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