Computational mathematical modeling of the nonlinear vibration characteristics of AFG truncated conical nano pipe based on the nonlocal strain gradient theory

In the present paper, the numerical dynamic analysis of a functionally graded nano-scale nonuniform tube was investigated according to the high-order beam theory coupled with the nonlocal gradient strain theory. The supposed crosssection is changed along the pipe length, and the material distribution, which combines both metal and ceramics, is smoothly changed in the pipe length direction, which is called axially functionally graded (AFG) pipe. Moreover, the porosity voids are dispersed in the cross-section and the radial pattern that the existence of both material distribution along the tube length and porosity voids make a two-dimensional functionally graded (2D-FG) truncated conical pipe. On the basis of the Hamilton principle, the governing equations and the associated boundary conditions equations are derived, and then a numerical approach is applied to solve the obtained equations.