%0 Journal Article
%T Free vibrations of nonlocally elastic rods
%A D. Prikazchikov
%A E. Avdeichik
%A G. Mikhasev
%J Mathematics and Mechanics of Solids
%@ 1741-3028
%D 2019
%R 10.1177/1081286518785942
%X Several of the Eringen¡¯s nonlocal stress models, including two-phase and purely nonlocal integral models, along with the simplified differential model, are studied in the case of free longitudinal vibrations of a nanorod, for various types of boundary conditions. Assuming the exponential attenuation kernel in the nonlocal integral models, the integro-differential equation corresponding to the two-phase nonlocal model is reduced to a fourth-order differential equation with additional boundary conditions, taking into account nonlocal effects in the neighbourhood of the rod ends. Exact analytical and asymptotic solutions of boundary-value problems are constructed. Formulas for natural frequencies and associated modes found in the framework of the purely nonlocal model and its ¡®equivalent¡¯ differential analogue are also compared. A detailed analysis of solutions suggests that the purely nonlocal and differential models lead to ill-posed problems
%K Asymptotic method
%K free longitudinal vibrations
%K nanorod
%K natural frequencies
%K two-phase integral model
%K Eringen¡¯s nonlocal elasticity
%U https://journals.sagepub.com/doi/full/10.1177/1081286518785942