Free vibrations of nonlocally elastic rods

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 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