The effect of the lattice parameter of functionally graded materials on the stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the stress field near crack tips subjected to a uniform anti-plane shear loading is investigated by means of the non-local theory. The traditional concepts of the non-local theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with coordinate parallel to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surface. To solve the dual integral equations, the displacement on the crack surface is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at crack tips. The nonlocal elastic solutions yield a finite hoop stress at crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field depends on the crack length, the parameter describing the functionally graded materials and the lattice parameter of materials.