Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams

In the current study, a nonlocal strain gradient beam model with third-order distribution of shear deformation is established to explore the nonlinear vibration of axially-loaded multilayer functionally graded graphene platelet-reinforced composite (GPLRC) nanobeams in both of the prebuckling and postbuckling domains. The dispersion of graphene platelet (GPL) nanofillers changes layerwise based upon different functionally graded patterns while it remains constant within each individual layer. The effective mechanical properties of multilayer functionally graded GPLRC nanobeams are estimated using Halpin-Tsai model of micromechanics. Hamilton's principle is utilized to construct the size-dependent differential equations of motion. Subsequently, an improved perturbation technique in conjunction with the Galerkin method is employed to present explicit analytical expression for nonlocal strain gradient nonlinear frequency in terms of applied axial load. It is observed that at the critical buckling point, the significance of the nonlocality and strain gradient size dependency on the nonlinear frequency remains constant for all values of maximum deflection. However, within the prebuckling and postbuckling regimes, by increasing the maximum deflection of the axially-loaded multilayer GPLRC nanobeam, both types of size effects on the nonlinear frequency reduce. Also, it is seen that similar to the type of GPL dispersion pattern, the value of GPL weight fraction has also no influence on the significance of size dependencies in the nonlinear frequency of axially-loaded multilayer functionally graded GPLRC nanobeams.