Mathematical models of nonlinear dynamics of functionally graded nano/micro/macro-scale porous closed cylindrical Kirchhoff–Love shells

The article presents new mathematical models for the dynamics of nonlinear nano/micro/macro-scale functionally graded porous closed cylindrical shells. The Kirchhoff–Love hypothesis is chosen as the kinematic model for the shells. Geometric nonlinearity is considered according to the von Karman model. Nanoeffects are accounted for using by a modified moment theory of elasticity. Variational and differential equations, as well as boundary and initial conditions, are derived from Hamilton's principle. A proof of the existence of a solution is conducted based on the theory of generalized solutions to differential equations (using methods of Hilbert spaces and variational methods).
As examples, nano/micro/macro-scale closed cylindrical shells are considered as systems with “almost” an infinite number of degrees of freedom subjected to banded transverse alternating loading. The Bubnov–Galerkin method in higher approximations is adopted as the method for reducing partial differential equations to the Cauchy problem. Its convergence is investigated.
The Cauchy problem is solved using Runge–Kutta methods of fourth to eighth order accuracy and the Newmark method. The application of several numerical methods at each stage of modeling is necessary to ensure the reliability of the obtained results. The study of complex oscillation characteristics of the closed cylindrical nano/micro/macro-scale shell is conducted using nonlinear dynamics methods, which involve constructing signals, phase portraits, applying Fourier analysis, and various wavelet transformations, among which the Morlet wavelet proved to be the most informative.
An analysis of the type of chaotic oscillations is carried out based on the spectrum of Lyapunov exponents using the Sano–Sawada method and the dominant exponent through several methods: Kanca, Rosenstein, and Wolf. It is shown that the size-dependent parameter and the consideration of porosity have a significant impact on the nature of the oscillations of cylindrical shells. The phenomenon of hyper-chaos has been discovered.