A variational boundary integral method for the analysis of three-dimensional cracks of arbitrary geometry in anisotropic elastic solids

A variational boundary integral method is developed for the analysis of three-dimensional cracks of arbitrary geometry in general anisotropic elastic solids. The crack is modeled as a continuous distribution of dislocation loops. The elastic energy of the solid is obtained from the known expression of the interaction energy of a pair of dislocation loops. The crack-opening displacements, which are related to the geometry of loops and their Burgers vectors, are then determined by minimizing the corresponding potential energy of the solid. In contrast to previous methods, this approach results in the symmetric system of equations with milder singularities of the type 1/R, which facilitate their numerical treatment. By employing six-noded triangular elements and displacing midside nodes to quarter-point positions, the opening profile near the front is endowed with the accurate asymptotic behavior. This enables the direct computation of stress intensity factors from the opening displacements. The performance of the method is assessed by the analysis of an elliptical crack in the transversely isotropic solid. It also illustrates that the conventional average schemes of elastic constants furnish quite inaccurate results when the material is significantly anisotropic.