Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

The nonlinear free vibration of simply supported porous functionally graded (PFG) size-dependent tubes under compressive axial loading in pre-buckling regime are investigated. The nonlinear Euler–Bernoulli beam hypothesis is employed within the framework of the nonlocal strain and velocity gradient theory to derive the nonlinear partial differential equation of motion. It is assumed that the material properties are gradually graded in the radial direction. Additionally, two different porosity distribution patterns are used in the radial direction. The modified power-law function is employed to estimate the effective properties of PFGM/NTs. The partial differential equation governing the lateral nonlinear vibration of prestressed PFGM/NTs is achieved by employing Hamilton’s principle. The method of multiple scales is utilized to solve the nonlinear ordinary differential equation system obtained by applying the Galerkin method to the nonlinear partial differential equation of lateral motion. The nonlinear frequencies of pre-buckled PFGM/NTs, which reflect the effects of the amplitude of different modes involved in vibration response, are formulated. The effects of different parameters, such as length scale parameters, porosity distribution pattern, and material composition, on the nonlinear frequencies of pre-buckled porous functionally graded size-dependent tubes are examined. The pre-buckling behavior stage demonstrates that porosity reduces the static stability of FGM/NTs, but the power-law index value and modulus of elasticity ratio can modify the softening effects of porosity. Research indicates that the presence and distribution pattern of porosity, along with material composition, can enhance the nonlinear frequencies of M/NTs.