Nonlinear modal analysis of multi-walled nanotube oscillations using nonlocal anisotropic elastic shell model

Abstract

System of nonlinear partial differential equations, which describes the multi-walled carbon nanotube nonlinear oscillations, is derived. The Sanders-Koiter nonlinear shell theory and the nonlocal anisotropic Hooke’s law are used in this model. Three kinds of nonlinearities are accounted. First of all, the van der Waals forces are nonlinear functions of the radial displacements. Secondly, the nanotube walls displacements have moderate values, which are described by the geometrically nonlinear shell theory. Thirdly, as the stress resultants are the nonlinear functions of the displacements, the additional nonlinear terms in the equations of motions are obtained. These terms are derived from the natural boundary conditions, which are used in the weighted residual method. The finite degrees of freedom nonlinear dynamical system is derived to describe the oscillations of nanostructure. The Shaw-Pierre nonlinear normal modes in the form of the multi-mode invariant manifolds are used to describe the free nonlinear oscillations, as the dynamical systems contains the internal resonances 1:1. The motions on the invariant manifolds are described by two degrees of freedom nonlinear dynamical systems, which are analyzed by the multiple scales method. The backbone curves of the nonlinear modes are analyzed. As follows from the results of the numerical simulations, the eigenmode of low eigenfrequency has commensurable longitudinal, transversal and circumference displacements. The nonlinear parts of the van der Waals forces harden essentially the backbone curve of the oscillations close to this eigenmode.