Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity
DOI:
https://doi.org/10.21638/spbu01.2021.203Abstract
Free high-frequency longitudinal vibrations of an inhomogeneous nanosized rod are studied on the basis of the nonlocal theory of elasticity. The upper part of spectrum with the wavelength comparable to the internal characteristic dimension of a nanorod is examined. An equations in the integral form with the Helmholtz kernel, incorporating both local and nonlocal phases, is used as the constitutive one. The original integro-differential equation is reduced to the forth-order differential equation with variable coefficients, the pair of additional boundary conditions being deduced. UsingWKB-method, a solution of the boundaryvalue problem is constructed 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 which allows estimating the upper part of spectrum of eigen-frequencies. Considering the nanorod with a variable cross-section area, we revealed a fair convergence of eigen-frequencies found in the framework of two models when the local fraction in the two-phase model vanishes.Keywords:
nanosized inhomogeneous rod, high-frequency vibrations, two-phase nonlocal theory of elasticity, asymptotic method
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Литература
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References
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2. Andrianov I.V., Awrejcewicz J., Weichert D. Improved continuous models for discrete media. Mathematical Problems in Engineering 2010, 986242 (2009). https://doi.org/10.1155/2010/986242
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6. Romano G., Barretta R., Diaco M., de Sciarra F.M. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Engineering Science 121, 151–156 (2017).
7. Mikhasev G., Nobili A. On the solution of the purely nonlocal theory of beamelasticity as a limiting case of the two-phase theory. International Journal of Solids and Structures 190, 47–57 (2020).
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Published
2021-07-21
How to Cite
Mikhasev, G. I. (2021). Studying free hight-frequency vibrations of an inhomogeneous nanorod based on the nonlocal theory of elasticity. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(2), 220–232. https://doi.org/10.21638/spbu01.2021.203
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In memoriam of P. E. Tovstik
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
