Investigation of the behavior of a mode-I crack in functionally graded materials by non-local theory

In this paper, the behavior of a mode-I crack in functionally graded materials is investigated by means of the non-local theory. The traditional concepts of the non-local theory are firstly extended to solve the mode-I crack fracture problem in functionally graded materials, in which the shear modulus varies exponentially with coordinate parallel to the crack. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are jumps of displacements across crack surfaces, not the dislocation density functions or the analysis functions. To solve the dual integral equations, the jumps of displacements across crack surfaces are directly expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The non-local elastic solutions yield a finite stress at crack tips, thus allowing us to use the maximum stress as a fracture criterion. Numerical examples are provided to show the effects of the crack length, the parameter describing functionally graded materials, the lattice parameter of materials and the material constants upon the stress fields near crack tips.