Perturbations for vibration of nano-beams of local/nonlocal mixture

Perturbations for vibration of nano-beams of local/nonlocal mixture

Ugurcan Eroglu, Giuseppe Ruta

download PDF

Abstract. Here we extend the perturbation approach, previously presented in the literature for Eringen’s two-phase local/nonlocal mixture model, to free vibration of purely flexible beams. In particular, we expand the eigenvalues and the eigenvectors into power series of the fraction coefficient of the non-local material response up to 2nd order. We show that the family of 0th order bending couples satisfy the natural and essential boundary conditions of the 1st order; hence, the 1st order solution can conveniently be constructed using the eigenspace of the 0th order with no necessity of additional conditions. We obtain the condition of solvability that provides the incremental eigenvalue in closed form. We further demonstrate that the 1st order increment of the eigenvalue is always negative, providing the well-known softening effect of long-range interactions among the material points of a continuum modelled with Eringen’s theory. We examine a simply supported beam as a benchmark problem and present the incremental eigenvalues in closed form.

Perturbation, Nonlocal Elasticity, Free Vibration

Published online 3/17/2022, 6 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA

Citation: Ugurcan Eroglu, Giuseppe Ruta, Perturbations for vibration of nano-beams of local/nonlocal mixture, Materials Research Proceedings, Vol. 26, pp 619-624, 2023


The article was published as article 100 of the book Theoretical and Applied Mechanics

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

[1] Eringen A.C., “Linear theory of nonlocal elasticity and dispersion of plane waves”, International Journal of Engineering Sciences 10 (1972), 425–435.
[2] Eringen A.C., Edelen D., “On nonlocal elasticity”, International Journal of Engineering Sciences 10 (1972), 233–248.
[3] Romano G., Barretta R., Diaco M., Marotti de Sciarra F., “Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams”, International Journal of Mechanical Sciences 121 (2017), 151–156.
[3] Eringen A.C., Nonlocal Continuum Field Theories. Springer (2002).
[4] Eroglu U., “Perturbation approach to Eringen’s local/nonlocal constitutive equation with applications to 1-D structures”, Meccanica, 55 (2020), 1119-1134.
[5] Fernández-Sáez J., Zaera R., Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory, International Journal of Engineering Science 119 (2017), 232–248.
[6] Fakher M., Hosseini-Hashemi S., Nonlinear vibration analysis of two-phase local/nonlocal nanobeams with size-dependent nonlinearity by using Galërkin method, Journal of Vibration and Control 27(3-4) (2021), 378–391.