The article deals with the large‐amplitude vibration of carbon nanotube‐based double‐curved shallow shells (CNTBDCSSs). After mathematical modeling of CNTBDCSSs, the von Karman‐type nonlinear basic relations of CNTBDCSSs are created, and then the nonlinear equations of motion are derived. Using an extended mixing rule, the CNTBDCSSs are estimated approximately by introducing some performance parameters. Four different carbon nanotube distributions are considered. The nonlinear basic equations are solved applying superposition, Galerkin, and semi‐inverse methods; and the frequency–amplitude relation for the large‐amplitude vibration of CNTBDCSSs is obtained. The nonlinear frequency to linear frequency ratio of CNTBDCSSs is determined as a function of the amplitude. The results are compared with published results to check the reliability and accuracy of the proposed formulation. It follows a systematic investigation aimed at checking the sensitivity of the nonlinear response to some reinforcement parameters and geometry, as the distribution of CNTs within the matrix.

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In this work, we discuss the free vibration behavior of thin-walled composite shell structures reinforced with carbon nanotubes (CNTs) in a nonlinear setting and resting on a Winkler–Pasternak Foundation (WPF). The theoretical model and the differential equations associated with the problem account for different distributions of CNTs (with uniform or nonuniform linear patterns), together with the presence of an elastic foundation, and von-Karman type nonlinearities. The basic equations of the problem are solved by using the Galerkin and Grigolyuk methods, in order to determine the frequencies associated with linear and nonlinear free vibrations. The reliability of the proposed methodology is verified against further predictions from the literature. Then, we examine the model for the sensitivity of the vibration response to different input parameters, such as the mechanical properties of the soil, or the nonlinearities and distributions of the reinforcing CNT phase, as useful for design purposes and benchmark solutions for more complicated computational studies on the topic.

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This study presents the solution for the thermal buckling problem of moderately thick laminated conical shells consisting of carbon nanotube (CNT) originating layers. It is assumed that the laminated truncated-conical shell is subjected to uniform temperature rise. The Donnell-type shell theory is used to derive the governing equations, and the Galerkin method is used to find the expression for the buckling temperature in the framework of shear deformation theories (STs). Different transverse shear stress functions, such as the parabolic transverse shear stress (Par-TSS), cosine-hyperbolic shear stress (Cos-Hyp-TSS), and uniform shear stress (U-TSS) functions are used in the analysis part. After validation of the formulation with respect to the existing literature, several parametric studies are carried out to investigate the influences of CNT patterns, number and arrangement of the layers on the uniform buckling temperature (UBT) using various transverse shear stress functions, and classical shell theory (CT).

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.

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