Keele Research Repository

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.