Dynamic stability of porous functionally graded nanotubes via nonlocal strain and velocity gradient theory

Dynamic stability of simply supported porous functionally graded (PFG) nanotubes under parametric axial excitation is investigated. The material properties which vary radially are described according to a power law function including the volume fraction of porosity. The nonlocal strain and velocity gradient theory together with Euler–Bernoulli hypothesis is employed to incorporate the size effects into the governing partial differential equation of motion. To discretize the considered continuum, the Galerkin approach is used. To investigate the effects of length scale parameters (i.e., nonlocal nanoscale parameter, gradient length scale, and inertia parameter) as well as power low index and porosity volume fraction on the instability region, Bolotine's method is utilized. Free vibration and buckling characteristics of PFG nanotubes are discussed as well. Findings show the opposite effects of the gradient length scale parameter and inertia parameter on natural frequencies and instability region of PFG nanotubes. It is seen that with an increase in the length scale parameter or decrease in the nonlocal nanoscale parameter or velocity gradient length scale parameter, the instability region will be wider and the corresponding excitation frequency grows.