On the flexoelectric effect on size-dependent static and free vibration responses of functionally graded piezo-flexoelectric cylindrical shells

Abstract

The flexoelectric effect depends on the strain gradient, which naturally appears in intelligent structures at micro and nano scales. Therefore, the flexoelectric phenomenon significantly affects electro-mechanical behavior of structures in these scales. In this study, the first-order shear deformation theory and the reformulated flexoelectric theory are applied to derive coupling equations of motion for a size-dependent static and dynamic responses of multilayered composite shell with simply supported and clamped edges. Constitutive equations are defined based on assumption that the internal energy function depends not only on strains and polarization tensors but also on strains and polarization gradient tensors. The piezo-flexoelectric structure is made of an isotropic functionally graded core and two orthotropic layers. Reynolds transport theorem and variational Hamilton principle are used to derive equations of motion expressed by forces and displacements, and corresponding boundary conditions. In order to check the correctness of the formulation, and solution of the problem obtained by Galerkin method, static and free vibrations analyses of macro-/micro-/nanoshell are investigated. Diagrams of deflection and electrical potential variations of the structure under the load are obtained for the direct flexoelectric effect. Additionally, the influence of various parameters, e.g., geometric properties, size effect, power-law index, and flexoelectric layer thickness, on the deflection, potential function, and natural frequency of the microshell have been investigated. The numerical results and discussion present that above parameters significantly affect a mechanical behavior of the microshell under consideration.