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

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 2^nd order. We show that the family of 0^th order bending couples satisfy the natural and essential boundary conditions of the 1^st order; hence, the 1^st order solution can conveniently be constructed using the eigenspace of the 0^th 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 1^st 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.