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Free high-frequency vibrations of nonlocally elastic beam with varying cross-section area

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Abstract

In the framework of the two-phase theory of nonlocal elasticity we study free high-frequency vibrations of nanobeams with varying cross section. Considering the Helmholtz kernel in the constitutive equation, we reduce the original integro-differential equation to an equivalent six-order differential equation with variable coefficients and derive the pair of additional boundary conditions accounting for the nonlocal edge effects. Applying WKB method, a solution of the boundary-value problem is constructed in the form of superposition of the outer expansion and edge effect integrals with high variability along the beam axis. As an example, a clamped nanobeam with a shape matching to the shape of a slab waveguide in a photonic crystal beam laser is considered. The derived relations for natural frequencies and performed calculations revealed strong effect of both the nonlocal model fraction and the cross-section variability on the correction to natural frequencies calculated within the classical beam theory

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  1. Department of Bio- and Nanomechanics, Faculty of Mechanics and Mathematics, Belarusian State University, Nezavisimosty Ave., 4, 220030, Minsk, Belarus

    Gennadi Mikhasev

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Correspondence to Gennadi Mikhasev.

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Communicated by Marcus Aßmus, Victor A. Eremeyev and Andreas Öchsner.

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The author acknowledges the support from the State Program of Scientific Investigations in Belarus “Convergence”.

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Mikhasev, G. Free high-frequency vibrations of nonlocally elastic beam with varying cross-section area. Continuum Mech. Thermodyn. 33, 1299–1312 (2021). https://doi.org/10.1007/s00161-021-00977-6

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