Free vibration of FG-CNT reinforced composite spherical shell panels using Gram-Schmidt shape functions

Free vibration characteristics of carbon nanotube reinforced composite spherical panels are studied in the present research. First order shear deformation shell theory and the Sanders kinematics are considered as the basic assumptions. Distribution of carbon nanotubes (CNTs) across the panel thickness may be uniform or functionally graded. Equivalent properties of the media are estimated according to a modified rule of mixtures approach which consists efficiency parameters to capture the size dependency of the properties. Using Hamilton's principle and the conventional Ritz formulation, the matrix representation of the equations associated with the free vibration motion is obtained. Shape functions of the Ritz method are obtained according to the Gram-Schmidt process. The resulting eigenvalue problem is solved to obtain the frequencies as well as mode-shapes of the spherical panel reinforced with CNTs. Convergence and comparison studies are provided to assure the effectiveness and accuracy of the proposed method. Afterwards, parametric studies are given to explore the effects of volume fraction of CNTs, distribution pattern of CNT, boundary conditions and geometric characteristics of the panel. It is shown that, enrichment of the polymeric matrix with more CNT results in higher frequencies. Furthermore, graded pattern of CNT is an influential factor on frequencies.