Nonlinear forced vibration analysis of functionally graded nanobeams in thermal environments with considering surface stress and nonlocal effects

In this investigation, an exact solution is proposed for the nonlinear forced vibration analysis of nanobeams made of functionally graded materials (FGMs) in thermal environment with considering the effects of surface stress and nonlocal elasticity theory. The physical properties of FGM nanobeams are assumed to vary through the thickness direction on the basis of the power law distribution. The geometrically nonlinear equations of motion and corresponding boundary conditions are derived using Hamilton’s principle on the basis of the Euler-Bernoulli beam theory. Using the Gurtin-Murdoch and Eringen elasticity theories, the surface stress and nonlocal effects are taken into account in obtained equations, respectively. For the solution purpose, first, the Galerkin procedure is employed in order to reduce the nonlinear partial differential governing equation into a nonlinear ordinary differential equation. This new equation is solved analytically by the multiple scales perturbation method. In the results section, the influences of different parameters including power law index, surface stress, nonlocal parameter, boundary conditions and temperature changes on the nonlinear forced vibration response of nanobeams are investigated. Also, comparisons are made between the results obtained from the classical, Gurtin-Murdoch and Eringen elasticity theories. It is shown that as the thickness decreases, the surface stress effect moderates the hardening-type nonlinear behavior of nanobeams. This effect is more prominent at low magnitudes of thickness. Moreover, one can find that by increasing the nonlocal parameter, the hardening-type response of nanobeams is intensified.