Asymptotic DQ solutions of functionally graded annular spherical shells

Abstract

In conjunction with the methods of differential quadrature (DQ) and asymptotic expansion, the three-dimensional (3D) solution for the static analysis of functionally graded annular spherical shells is presented. The materials are assumed to be isotropic elastic and inhomogeneous through the thickness direction. By means of proper nondimensionalization and then asymptotic expansion to the field variables, we decompose the basic 3D equations in a series of differential equations for various orders. Upon successively integrating these resulting equations through the thickness direction, we further obtain the recursive sets of governing equations for various orders. The edge boundary conditions at each order level are derived as their resultant forms. The governing equations at each order level associated with the corresponding edge boundary conditions will compose a well-posed boundary-value problem. The DQ method is adopted for solving the boundary-value problems for various orders. Since the differential operators of the governing equations remain the same for various orders and the nonhomogeneous terms at the higher-order level can be calculated from the lower-order solution, the solution procedure for the leading order problem can be repeatedly applied for the higher-order problems. The functionally graded annular spherical shells with simply supported edges subjected to sinusoidally and uniformly distributed loads are considered in the illustrative examples. The convergence and accuracy of the present asymptotic DQ solutions are examined.