Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects

Abstract

Recently, advanced materials whose properties vary within a continuous pattern have been put to use to design and manufacture modern structures. In the current investigation, size dependencies are captured in the nonlinear free vibration characteristics of micro/nano-beams made of bi-directional functionally graded materials (2D-FGM). With the aid of the nonlocal strain gradient elasticity theory and the variational principle, the size-dependent nonlinear differential equations of motion are derived within the framework of the refined hyperbolic shear deformation beam theory. It is supposed that the material properties are distributed exponentially along longitudinal direction, and vary based on the power law function in lateral direction. Moreover, the deviation of the associated physical neutral plane from the mid-plane counterpart is taken into consideration. By employing a numerical solution methodology on the basis of the generalized differential quadrature method (GDQM) together with Galerkin technique and pseudo arc-length continuation method, the nonlocal strain gradient frequency-deflection responses of 2D-FGM micro/nano-beam are obtained corresponding to various values of longitudinal and lateral material property indexes and small scale parameters. It is revealed that the increment made by the strain gradient size dependency in the value of the nonlinear frequency is more than the reduction caused by the nonlocality, especially for the lower maximum deflection imposed to the 2D-FGM micro/nano-beam. Also, it is indicated that for lower values of the material property gradient indexes, the reduction in the nonlinear frequency caused by the lateral functionally graded pattern in the absence of the axial functionally graded pattern is more than that made by the vice versa case. However, for higher values of the material property gradient indexes, an opposite observation is seen.