Fracture mechanics analysis of functionally graded materials using a mixed collocation element differential method

In this paper, the fracture mechanics analysis in functionally graded materials and structures (FGMs) is presented. The elemental differential method, is extended to simulate the fracture behaviors of the functionally graded materials, in which the system of equations is established directly based on the equilibrium equations. The first and second order differentiations of the shape functions are utilized to interpolate the geometrical and physical variables within the isoparametric elements. A novel collocation strategy is introduced to construct the system of equations by the governing equations and the traction equilibrium equations according to the nodal distributions in the mesh grids of the structures. Furthermore, a mixed collocation element differential method is further proposed to handle the singular points in the computation domains such as the crack tips and structural corners. The weak-form formulations, such as the weighted residuals approach, are utilized to establish the system of equations for nodes within the domain of elements. Thus, the strong-weak form method can combine the superiorities of the standard finite element methods and the strong-form methods for the aspects of easily constructing shape functions and directly generating system of equations. Numerical examples about the stress intensity factors of the static and dynamic problems in functionally graded materials are presented to validate the proposed methods.