On the nonlinear dynamics of a multi-scale flexoelectric cylindrical shell

Abstract

The flexoelectric property shows the relationship between the strain gradient and the electric polarization and has a significant effect on static and dynamic responses of structures at small scales such as micro and nano. Nowadays, experimental studies for intelligent/composite ultra-small mechanical systems are complex, challenging, and in most cases still impossible, especially for nonlinear dynamic behavior analysis. For the first time, this study presents an advanced and complex generalized model for the nonlinear vibrations of a piezo-flexoelectric mechanical system based on a cylindrical shell. It should be noted, that the formulated boundary value problem is generalized, well-posed, and can be applied to the system at micro and nano scales. Hamilton's variational principle as well as assumptions of the first-order shear deformation theory (FSDT) and reformulated flexoelectric theory were used to derive equations of motion of the multi-scale intelligent shell system. To introduce nonlinear effects, von-Kármán strains are applied. Appropriate classical and non-classical mechanical/electric boundary conditions as well as higher-order forces and moments are determined to fully close the formulated problem. The Galerkin method combined with the GDQM, supported by the displacement control strategy and the weighted residual method, have been used to discretize and solve the nonlinear governing equations of the shell with different boundary conditions. Good convergence between the current results and the results from previous studies for simpler cases of the shell was proved. Furthermore, the results showed that geometric ratios of the structure, gradation of material properties, thickness of the flexoelectric layers, and the size effect parameter significantly influence the nonlinear frequency of the structure under the direct flexoelectric effect.