Nonlinear dynamics of rectangular nano-shells

Abstract

Mathematical model of non-linear vibrations of shallow, elastic, isotropic nano-shells with rectangular base subjected to transverse sign-variable load are constructed. Based on Kirchhoff-Love thin shell theory with von Kármán nonlinear strains and the modified couple stress theory (MCST), size-dependent governing equations and corresponding boundary conditions are established through Hamilton’s principle. The governing PDEs are reduced to ODEs by the second-order Finite Difference Method (FDM). The obtained system of equations is solved by Runge–Kutta methods of second order accuracy. The Cauchy problem is solved by the Runge–Kutta fourth-order method. We analyzed the convergence of these solutions depending on the step of integration over time and spatial coordinate. It was revealed, that taking into account nano-effects increases area of harmonic vibrations and leads to the appearance of as chaotic and hyperchaotic vibrations. The carried out numerical experiment shows, that the transition of vibrations from harmonic to chaotic follows to Feigenbaum’s scenario. In particular, to analyze the character type of vibration computation of largest Lyapunov exponents are employed. We found that hyperchaotic vibrations are characterized by two positive Lyapunov exponents and chaotic vibration by one positive Lyapunov exponent. For nano-shells, this phenomenon was discovered for the first time. Lyapunov exponents spectra estimated by different algorithms, including Wolf’s, Rosenstein’s, Kantz’s, and Sawada. Numerical examples of the theoretical investigations are given.