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For the first time, the nonlinear bending behavior of porous functionally graded (FG) curved nanotubes is studied in this paper. The stiffness enhancement and stiffness reduction effects are described by the nonlocal strain gradient theory. The FG curved nanotubes have uniformly distributed pores in the radial direction. The material properties of the nanotubes corresponds to a modified power-law function. The asymptotic solutions of the curved nanotubes is obtained by using the two-step perturbation method. It shows that the nonlinear bending behavior of the curved nanotubes is influenced by the size effects, power law index, porosity distribution, temperature variation, boundary conditions, and physical dimension. Furthermore, the jump changes as well as snap-through buckling can take place when the FG curved nanotubes are under the influence of normal bending loads.

In this study, the vibration behaviors of porous nanotubes are investigated for the first time. The nonlocal strain gradient theory in conjunction with a refined beam model are employed to formulate the size-dependent model. It is presumed that the porous nanotubes are made from functionally graded materials, and the material parameters of nanotubes relate to temperature variation and vary continuously in the radial direction. Employing a refined beam theory which includes the effects of transverse shear deformation, the equations of motion are derived based on Hamilton's variation principle and solved by the Navier solution method. Some comparisons are presented to valid the correctness of present solution method. The effects of the nonlocal parameter, strain gradient parameter, temperature variations, porosity volume fraction and material variation on the vibration characteristic of the nanotubes are discussed in detail.

For the first time, the size-dependent thermal buckling and post-buckling behavior of nanotubes made of functionally graded materials (FGMs) with porosities is investigated by using a refined beam theory. This non-classical nanotube model is based on Eringen nonlocal elasticity model which incorporates the small scale effect. Two types of porosity distribution, including even and uneven distribution, are taken into account. The material properties of the nanotubes are temperature-dependent and vary in the radial direction. The size-dependent governing differential equations are derived by employing the generalized variation principle and solved by using a two-step perturbation method. The effects of small scale parameter, porosity volume fraction, the volume fraction index and boundary conditions on thermal buckling and post-buckling of FGM nanotubes are studied by several numerical examples. It can be concluded that the porosity volume fraction and small scale parameter change the buckling and post-buckling behavior of the nanotubes.