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The present study establishes the nonlinear free vibration analysis of double-functional gradient carbon nanotubes-reinforced (DFG-CNTR) conical shells resting on elastic foundations. Carbon nanotubes are distributed with three different patterns along the thickness of the conical shell. The matrix material of the shell is considered to be a combination of metal and ceramic whose properties change as a power law function along the thickness of the shell. Firstly, the nonlinear dynamic equations of the DFG-CNTR conical shell are derived based on the first-order shear deformation theory (FOSDT) and von Karman’s strain-displacement relations. Then, with the help of the Galerkin discretization method, partial differential equations of the system are converted into time-dependent ordinary differential equations. The homotopy perturbation method is used to solve the nonlinear dynamical equation. Finally, a parametric study is presented to investigate the effects of some parameters of the system, such as the functionally graded material (FGM) power index, volume fraction and distribution pattern of carbon nanotubes, elastic foundations, boundary conditions and the geometric characteristics of the shell on the backbone curves of the DFG-CNTR conical shell.
Youseftabar et al. (Tue,) studied this question.