Nonlinear vibration of different types of functionally graded nanotubes using nonlocal strain gradient theory

Authors in this work investigate nonlinear vibration of different types of functionally graded tube in the theoretical framework of nonlocal strain graded theory. They are tubes with inside-out functionally graded distribution, tubes with circumferential functionally graded distribution and tubes with sinusoidal functionally graded distribution. At the beginning, based on the assumptions and appropriate mathematical models proposed via us, the effective material properties of new functionally graded materials are defined in the form of power-law. Then, a high order shear deformation beam model that can satisfy the stress boundary conditions on inner and outer surfaces is used to analyze respective types of FGM tube. With the aid of the Hamilton' principle, the nonlinear governing equations that include a nonlocal parameter and a material length-scale parameter are acquired, and then are solved via the perturbation method to obtain appropriate analytical solutions. Finally, parametric studies are carried out in detail for three types of functionally graded tube, including the thickness of tube, temperature, nonlocal parameter, strain gradient parameter, scale parameter ratio, slenderness ratio, volume index, different beam models, different kinds of functionally graded distribution and the size of lead