Mathematical modeling and solution of nonlinear vibration problem of laminated plates with CNT originating layers interacting with two-parameter elastic foundation

Generalizing the first-order shear deformation plate theory (FOPT) proposed by Ambartsumyan (Theory of anisotropic plates, Nauka, Moscow, 1967 (in Russian)) to the heterogeneous laminated nanocomposite plates and the nonlinear vibration problem is analytically solved taking into account an elastic medium in this study for the first time. The Pasternak-type elastic foundation model (PT-EF) is used as the elastic medium model. After creating the mathematical models of laminated rectangular plates with CNT originating layers on the PT-EF, the large amplitude stress-strain relationships and motion equations are derived in the form of nonlinear partial differential equations (PDEs) within FOPT. Then, by applying Galerkin's method to the derived equations, it is reduced to a nonlinear ordinary differential equation (NL-ODE) containing the second- and third-order nonlinear terms of the deflection function for laminated rectangular plates composed of nanocomposite layers. The NL-ODE is solved by the semi-inverse method, and the nonlinear frequency-amplitude relationship for the laminated plates consisting of CNT originating layers resting on the PT-EF is established within FOPT for the first time. From these relations, similar relations can be obtained particularly for the unconstrained laminated and monolayer CNT patterns plates. After comparing the accuracy of the obtained formulas with the reliable results in the literature, comprehensive numerical analyses are performed.

Automated summary

In this study, the first-order shear deformation plate theory (FOPT) proposed by Ambartsumyan is extended to heterogeneous laminated nanocomposite plates and the nonlinear vibration problem. An elastic medium is considered for the first time, using the Pasternak-type elastic foundation model (PT-EF). The mathematical models of laminated rectangular plates with CNT originating layers on the PT-EF are created, and the large amplitude stress-strain relationships and motion equations are derived within FOPT. By applying Galerkin's method, the derived equations are reduced to a nonlinear ordinary differential equation (NL-ODE) for laminated rectangular plates composed of nanocomposite layers. The NL-ODE is solved by the semi-inverse method, and the nonlinear frequency-amplitude relationship is established for the laminated plates resting on the PT-EF within FOPT for the first time. From these relations, similar relations for unconstrained laminated and monolayer CNT patterns plates can be obtained.