On vibration of functionally graded nano-tubes in the thermal environment

In this paper, attention is paid to the prediction of vibration characteristics of two-dimensional functionally graded (2D-FG) tubes based on higher order theory. The tubes are formed of two different materials, and the material properties of the nano-scaled tubes vary both in the length and radial direction. To incorporate the size effect, the nonlocal strain gradient theory is employed. The thermal force is considered to be uniform temperature across the radial of the nonlocal tube, on the basis of the Hamilton's principle, the size-dependent governing equations are derived. After solving these equations using the generalized differential quadrature method (GDQM), the impact of the nonlocal parameter, strain gradient parameter, temperature variations, material variation on the vibration characteristic of the tubes with various boundary conditions are discussed in details.