Vibrational analysis of functionally graded carbon nanotube-reinforced composite spherical shells resting on elastic foundation using the variational differential quadrature method

In this paper, the free vibration characteristics of embedded functionally graded carbon nanotube-reinforced composite (FG-CNTRC) spherical shells are studied based on a numerical approach. The elastic foundation is considered to be Pasternak-type. Moreover, the extended rule of mixture is used so as to obtain the material properties of FG-CNTRC. The shell is also modeled according to the first-order shear deformation shell theory. The energy functional of the structure is obtained first. Using differential operators, the discretized form of the energy functional is derived. By means of the variational differential quadrature (VDQ) method, the reduced forms of mass and stiffness matrices are then obtained. Selected numerical results are given to investigate the effects of different parameters such as elastic foundation coefficients, boundary conditions, CNT volume fraction, thickness-to-radius ratio and type of distribution of CNT on the vibrations of FG-CNTRC spherical shells.