Three-dimensional vibration analysis of laminated functionally graded spherical shells with general boundary conditions

Abstract

Free vibration of laminated functionally graded (FG) spherical shells with general boundary conditions and arbitrary geometric parameters is studied in this paper. The study is based on the three-dimensional shell theory of elasticity and the energy based Rayleigh–Ritz procedure. It is assumed that the material properties of the laminated FG spherical shells vary continuously through the thickness direction according to power law distribution of the volume fraction of the constituents. Under the current framework, regardless of boundary conditions, each displacement variations of the laminated FG spherical shell is invariantly expanded as a modified Fourier series in which several supplementary terms are introduced to ensure and accelerate the convergence of the expansion and all the expanded coefficients are determined by the Rayleigh–Ritz procedure. The modified Fourier series results are presented and compared with the available accurate solutions to verify the validity of the current formulation. Detailed parametric investigation is carried out to examine the influences of boundary conditions, geometric parameters and material distributions on the natural frequencies of the spherical shells. Numerous vibration results for several laminated FG spherical shells with various boundary conditions are presented for different geometric parameters and power-law exponents, which may serve as benchmark solutions for future researches to evaluate the new 2-D shell theories and to compare results obtained by approximate numerical methods.