Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes

The aim of this paper is to numerically investigate the geometrically nonlinear primary resonance of third-order shear deformable functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates with various edge supports subjected to a harmonic excitation transverse force. The nanocomposite plates are composed of a mixture of matrix and single-walled carbon nanotubes (SWCNTs), and the effective mechanical properties are obtained by means of the rule of mixture. Employing the von Kármán hypotheses and Reddy's third-order shear deformation plate theory, the nonlinear equations of motion for the in-plane and out-of-plane directions as well as the corresponding boundary conditions are derived using Hamilton's principle. In the numerical solution procedure, the generalized differential quadrature (GDQ) method is used for the discretization, and a numerical Galerkin scheme is then employed to convert the discretized nonlinear partial differential equations (PDEs) into a Duffing-type nonlinear time-varying set of ordinary differential equations (ODEs). Afterwards, a time periodic discretization and the pseudo-arc length continuation method are utilized to determine the frequency- and force-responses of FG-CNTRC plates. Moreover, the curves corresponding to the nonlinear free vibration are provided. The influences of important parameters on the frequency-response and force-response curves of the FG-CNTRC plates are studied in the numerical results.