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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. • True meshless method is developed using Galerkin method to two dimensional elasticity. • System matrix can be obtained analytically for homogeneous rectangular sheets. • Discontinuous boundary value problem can be solved with one block. • All integral functions in domain and boundary integrals are regular and can be obtained easily. • Numerical procedure is convergent with very high accuracy. [ABSTRACT FROM AUTHOR]

• Erdogan fundamental solutions are employed in the method of fundamental solution for cracked body in functionally graded materials. • The accuracy and convergence of MFS have been observed; • Dynamic problem is solved without dynamic fundamental solutions; • Mapping method is applied in the domain integral. In this paper the Method of Fundamental Solutions (MFS) incorporating Erdogan's solutions for Functionally Graded Materials (FGM) is presented for analysis of 2D fracture problems subjected to static and dynamics loads. Erdogan derived analytical solutions for a pair of static concentrated force in an infinite isotropic plate with a straight cut. Based on homogenous isotropic analysis, the contribution from non-homogeneity in equilibrium equations is treated as body forces and domain integrals based on the Erdogan's fundamental solutions are required. In the domain integrals, all singularities are cancelled by introduction of a polar coordinate system at collocation points and crack tips. In the dynamic cases, the Laplace transformation with the Durbin inversion technique is adopted to determine the time-dependent variables such as the stress intensity factors at crack tips. The domain integrals are obtained numerically with the sub-region technique. The accuracy of the MFS is demonstrated with four numerical examples and comparisons are implemented with different numerical approaches. [ABSTRACT FROM AUTHOR]