Levy solution for bending response of FG carbon nanotube reinforced plates under uniform, linear, sinusoidal and exponential distributed loadings

Abstract

A new analytical approach for bending response of functionally graded single-walled carbon nanotube (FGCNT) reinforced plate resting on double-layered elastic foundations in thermal environments is presented based on Levy and Navier types solutions. The distribution of the carbon nanotubes is varied through the plate thickness in accordance with a modified power law. Four types of carbon nanotube distributions are considered. The four edges of the plate are simply supported for Navier method, whereas for Levy method, two opposite only of them are simply supported and the other ones are arbitrary. The present FGCNT plate is subjected to uniform, linear, sinusoidal or exponential distributed loadings. A refined shear deformation plate theory with four unknown functions is employed to obtain the closed form solution. The four coupled governing partial differential equations are derived utilizing Hamilton’s principle. Applying Levy solution and then the state space concept to the governing equations, a nonhomogeneous first-order linear ordinary differential system with constant coefficients is obtained. The solution of the homogeneous system (homogeneous solution) is obtained by using the matrix method. While, the method of undetermined coefficients is applied to find the particular solution of the nonhomogeneous linear system. The results obtained by Navier and Levy methods are compared with available results in the literature. Several examples are discussed for various values of the foundation stiffness, CNT volume fraction, various types of plate geometries, CNT distributions, external applied loads and boundary conditions.