A novel boundary element formulation for anisotropic fracture mechanics

A novel boundary element formulation for two-dimensional fracture mechanics is presented in this work. The formulation is based on the derivation of a supplementary boundary integral equation to be used in combination with the classic displacement boundary integral equation to solve anisotropic fracture mechanics problems via a single-region approach. The formulation is built starting from the observation that the displacement field for an anisotropic domain can be represented as the superposition of a vector field, whose components satisfy a suitably defined anisotropic Laplace equation, and the gradient of the Airy stress function. The supplementary boundary integral equation is then obtained using such representation into the integral expression of the aforementioned Laplace equation and employing the relationship between the stress function gradient and the boundary tractions. The supplementary equation neither requires the computation of hyper-singular integrals nor does it introduce additional variables for the problem, as it involves boundary displacements and tractions only. Numerical results are obtained for both uncracked and cracked bodies and show the accuracy and potential of the proposed approach.