A Study of Free High-Frequency Vibrations of an Inhomogeneous Nanorod, Based on the Nonlocal Theory of Elasticity

Abstract: Free high-frequency longitudinal vibrations of an inhomogeneous nanosize rod are studied on the basis of the nonlocal theory of elasticity. The upper part of the spectrum with a wavelength comparable to the internal characteristic dimension of the nanorod is investigated. An integral-form equation with a Helmholtz kernel, containing both local and nonlocal phases, is used as the constitutive one. The original integrodifferential equation is reduced to a fourth-order differential equation with variable coefficients and a pair of additional boundary-value conditions is obtained. Using the WKB-method, we construct a solution of the boundary-value problem in the form of the superposition of a main solution and edge-effect integrals. As an alternative model, we consider the purely nonlocal (one-phase) differential model, providing an estimate of the upper part of the spectrum of eigenfrequencies.