Size dependent free vibration analysis of multilayer functionally graded GPLRC microplates based on modified strain gradient theory

Abstract

In this study, for the first time, a size dependent computational model based on the modified strain gradient theory (MSGT) and higher-order shear deformation theory (HSDT) for free vibration analysis of multilayer functionally graded graphene platelet-reinforced composite (FG GPLRC) microplates is proposed. To capture size effects of microstructures, three material length scale parameters (MLSPs) are considered and used. The effective Young's modulus for each layer of the FG GPLRC microplate is computed according to the Halpin–Tsai model, while the effective density and Poisson's ratio are computed by the rule of mixtures. Four different types of graphene platelets (GPLs) distributions, which are either uniform or functionally graded (FG), are considered. The principle of virtual work is used to derive discretize governing equations which are then solved by an isogeometric analysis (IGA). In addition, thank to continuous higher-order derivatives of NURBS basis functions in IGA, it is suitable for a numerical implementation of the present size dependent model within required third-order derivatives in the weak form. Besides, the present size dependent model can be recuperated into the modified couple stress theory model (MCST) or classical HSDT model when two or all MLSPs in the theory are taken equal to zeros, respectively. The rectangular and circular FG GPLRC microplates with different boundary conditions, distributed types of GPLs and MLSPs are exampled to evaluate natural frequencies. Numerical results have shown that the difference of the natural frequency predicted by the MSGT, the MCST and classical HSDT are large when the plate thickness approaches the MLSPs, however, this difference decreases with a rise of the plate thickness.