Size-dependent parameter cancels chaotic vibrations of flexible shallow nano-shells

A theory of flexible shallow nano-shells is developed based on the modified couple stress theory in higher approximation. The shell material is considered as isotropic, elastic and both von Karman and Kirchhoff-Love hypotheses are taken into account. Variational Hamilton's principle yields differential equations of motion of both shallow size-dependent nano-shells with rectangular planforms and axially symmetric nano-shells with circular planforms. The derived PDEs are reduced to the Cauchy problem by using the FDM (finite difference method), and then solved by the Runge-Kutta-type methods. Convergence of the numerical results is analysed with respect to the number of the shell radius partitions and time step. The Fourier power spectra, various wavelets-type spectra, phase and modal portraits, as well as signs of the LEs (Lyapunov exponents) are investigated. All results associated with the analysis of LEs are validated based on the case studies of non-linear dynamics. Two kinds of boundary conditions are employed: movable and fixed clamping along the shell edge. It is shown that the size-dependent parameter essentially influences shell vibrations (in particular, the chaotic vibrations become periodic). (C) 2019 Elsevier Ltd. All rights reserved.