Nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite rectangular plates

The geometrically nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite (FG-GPLRC) rectangular plates with different edge conditions is examined. The considered plate with N-L-layers is made from a mixture of an isotropic polymer matrix and graphene platelets (GPLs) in each layer. The weight fraction of GPLs changes in a layer-wise manner. The modified Halpin-Tsai model and rule of mixture are utilized to compute the effective material properties of FG-GPLRCs. To mathematically model the vibrations of FG-GPLRC plates, the displacement field, strain tensor and constitutive relations as well as the energy functional of system including strain and kinetic energies and external work are represented in matrix forms as a function of the displacement components. Then, by simultaneous use of Hamilton's principle and an efficient numerical scheme namely, the variational differential quadrature (VDQ) technique, the weak form of discretized nonlinear equations of motion is obtained. The present model includes the influences of geometric nonlinearity, rotary inertia and transverse shear deformation. Furthermore, a multistep numerical approach based on the Galerkin method, time periodic discretization method and pseudo arc length continuation technique in conjunction with the modified Newton-Raphson method is employed to solve the problem of nonlinear harmonically excited vibration of FG-GPLRC rectangular plates. Results are plotted in the form of frequency-response and force-response curves to indicate the effect of various parameters such as GPL distribution pattern, weight fraction, geometry of GPL nanofillers and boundary constraints of FG-GPLRC plates.