Analysis of an arbitrarily oriented crack in a functionally graded plane using a non-local approach

This paper considers the problem of a mixed-mode crack embedded in an infinite graded medium in which the crack is arbitrarily oriented with respect to the material gradient. The material properties are assumed to have an exponential variation and the crack surfaces are subjected to both normal and tangential tractions which can be related to the external applied loads. Classical elasticity equations are reformulated to incorporate non-local effects by employing Eringen's non-local theory resulting in a non-singular stress field. Using Fourier transform, two integral equations are derived where the unknowns are the jumps in displacements across the crack surfaces. These jumps are expanded in a series of Jacobi polynomials and then substituted into the integral equations which are solved using the so-called Schmidt method. In contrast with classical elasticity solutions, there are no stress singularities at the crack tips in this solution owing to the incorporation of non-local effects. Therefore, the use of non-local theory permits the use of maximum stress in conjunction with a brittle fracture criterion. The principal objective of this study is to investigate the effect of various parameters such as crack length, material gradient nonhomogeneity parameter, angular orientation of the crack and lattice parameter on the non-local stress field near the crack tips.