Finite difference method for solving crack problems in a functionally graded material

A linear elastic two-dimensional formulation for functionally graded materials is presented. The two-dimensional equilibrium equations and boundary conditions in an orthogonal curvilinear coordinate system are written explicitly. The finite difference technique is used to solve the above formulation. The solution technique is verified by solving two test problems, in which the material is graded horizontally and vertically. The results are compared to analytical results and have very good agreement. The solution technique is then applied to solve a long layer containing an edge crack in which it is assumed that the Young's modulus varies continuously along its width. The problem is solved for two loading conditions: tension and bending. The mode I stress intensity factor is extracted by applying three methods: J line and two versions of a modified conservative J integral for graded materials. All three methods provide similar results, which are in excellent agreement with the semi-analytical results in the literature. These results demonstrate the applicability of the finite difference technique for solving crack problems in functionally graded materials.