On the computation of mixed-mode stress intensity factors in functionally graded materials

A new interaction energy integral method for the computation of mixed-mode stress intensity factors at the tips of arbitrarily oriented cracks in functionally graded materials is described. In the method, interaction energy contour integrals are defined and expressed in equivalent domain form. The interaction energy integrals involve products of the actual fields, that arise from solution to the boundary value problem, with known auxiliary fields. The auxiliary stress and displacement fields are chosen to be the asymptotic near-tip fields for a crack in a homogeneous material having the same elastic constants as those found at the crack tip in the functionally graded material. The auxiliary strain fields are obtained from the auxiliary stress fields using the constitutive relation for the functionally graded material. A consequence of our choice for the auxiliary strain field is lack of compatibility which leads to extra terms in the domain integrals that need to be evaluated for the sake of accuracy. The mixed-mode stress intensity factors are obtained from the domain integrals as a post-processing step in the extended finite element method. To assess the accuracy of the method, we consider the benchmark problems of an edge-cracked plate and an angled center crack for specimens with a functional gradient in material properties. Excellent agreement is obtained between the numerical results and the analytical solutions for both stress intensity factors in all cases. All numerical results for the stress intensity factors also exhibit domain independence. The pertinent post-processing routines are provided for download via the world wide web.