Nonlinear Vibration of a Functionally Graded Nanobeam Based on the Nonlocal Strain Gradient Theory considering Thickness Effect

In this work, a nonlocal strain gradient beam model considering the thickness effect is developed to study the nonlinear vibration response of a functionally graded nanobeam. The governing equation of the functionally graded nanobeam is derived by using the Euler–Bernoulli beam theory with von Kármán’s nonlinear strain‐gradient relationship and the Hamilton principle. The expression of the nonlinear frequency for the functionally graded nanobeam with pinned‐pinned boundary conditions is obtained with the help of Galerkin technique and the Hamiltonian approach. The obtained results show that the effect of thickness is very important for the size‐dependent vibration response of the functionally graded nanobeam; the nonlinear vibration response of the nanobeam depends not only on the material length scale parameter and nonlocal parameter but also on the slenderness ratio. Effects of the slenderness ratio and the power‐law index on the vibration response of the functionally graded nanobeam are also investigated and discussed. The numerical results show that the nonlocal parameter reduces the nonlinear frequency of the functionally graded nanobeam, while the material length scale parameter increases the nonlinear frequency of the functionally graded nanobeam. The slenderness ratio leads to an increase in the nonlinear frequency of the functionally graded nanobeam, while the power‐law index leads to a decrease in the nonlinear frequency of the functionally graded nanobeam.