Small Scale Effects on the Large Amplitude Nonlinear Vibrations of Multilayer Functionally Graded Composite Nanobeams Reinforced with Graphene-Nanoplatelets

   The main purpose of the present investigation is to analyze more comprehensively the size-dependent nonlinear free vibration response of multilayer functionally graded graphene platelet-reinforced composite (GPLRC) nanobeams. As a consequence, both of the hardening stiffness and softening stiffness of size effect are taken into consideration by implementation of the nonlocal strain gradient elasticity theory within the framework of the third-order shear deformation beam theory. The graphene platelet (GPL) nanofillers are dispersed uniformly or in accordance with three different functionally graded patterns based on a layerwise change of their weight fraction through the individual layers. Halpin-Tsai micromechanical scheme is utilized to estimate the effective material properties of multilayer functionally graded GPLRC nanobeams. With the aid of the Hamilton's principle, the non-classical governing differential equations of motion are derived. After that, an improved perturbation technique in conjunction with the Galerkin method is employed to achieve an explicit analytical solution for nonlocal strain gradient nonlinear frequency of multilayer functionally graded GPLRC nanobeams. It is indicated that at zero vibration amplitude, the pattern of GPL dispersion has no influence on the significance of the size dependencies. However, by taking the large vibration amplitude into account, both of the nonlocality and strain gradient size effects on the nonlinear frequency of O-GPLRC and X-GPLRC nanobeams are minimum and maximum, respectively.