Nonlinear free vibrations analysis of geometrically imperfect FG nano-beams based on stress-driven nonlocal elasticity with initial pretension force

Size-dependent flexural nonlinear free vibrations of geometrically imperfect straight Bernoulli-Euler functionally graded nano-beams are investigated by the stress-driven nonlocal integral model (SDM). By using the Galerkin method, the governing equations is reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency for Simply-Supported, Clamped-Simply Supported and Clamped-Clamped nano-beams is then established using the Hamiltonian approach to nonlinear oscillators. Effects of nonlocal scale parameter and an initial axial tension force on fundamental frequencies are examined and compared with those obtained by Eringen's nonlocal model. It is shown that the nonlinear approach based on nonlocal stress model, with the appropriate constitutive boundary conditions, is capable of capturing the dynamical responses of the nano-beams and provides an advantageous method for the design and the optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).