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

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. © 2023, Association of American Publishers. All rights reserved.