Galerkin finite block method in solid mechanics

Abstract

In this paper, the Galerkin Finite Block Method (GFBM) is proposed for the first time to solve two-dimensional elasticity problems with functionally graded materials (FGMs). The physical variables are approximated with double layer Chebyshev polynomials with unknown coefficients for two-dimensional elasticity. The main idea is to establish a set of linear algebraic equations by the Galerkin method using Chebyshev polynomials in the weak form of the partial differential equations and the boundary conditions. By introducing the mapping technique, a block with irregular shape of the boundary is transformed from the Cartesian coordinates (x, y) to the normalized coordinates (ξ, η), |ξ|⩽1, |η|⩽1. The continuity conditions of the displacement and traction on the interface between two blocks are introduced both in the weak form and strong form respectively. Several numerical examples are presented and comparisons are made with the Finite Element Method (FEM) including a cracked FGM sheet, in order to demonstrate the accuracy and efficiency of the GFBM.