Nonlinear normal modes of multi-walled nanoshells with consideration of surface effect and nonlocal elasticity

Abstract

System of nonlinear partial differential equations with respect to displacements is derived to describe free nonlinear vibrations of multi-walled nanoshell. The geometrically nonlinear vibrations of nanoshell, which is described by Sanders-Koiter nonlinear shell theory, are considered with account of nonlinear van der Waals forces, surface effect and nonlocal elasticity. Using the weighted residual method, the system of the partial differential equations is transformed into the system of nonlinear ordinary differential equations. The nonlinear boundary condition for the longitudinal stresses resultant is accounted in these differential equations.The essentially curvilinear Kauderer-Rosenberg nonlinear normal modes are studied numerically by combination of shooting technique and continuation algorithm. Such NNMs in double-walled nanoshell are undergone the Naimark- Sacker bifurcation. As a result, the curvilinear nonlinear modes are transformed into chaotic motions.