Summary: The application of the extended finite element method (XFEM) to fracture mechanics problems enables one to obtain accurate solutions more efficiently than with the standard finite element method. A component can be modelled without the need to build a mesh that matches the crack geometry, and thus remeshing as the crack grows is unnecessary. In the XFEM approach, the interpolation on certain elements is enriched with functions that make it feasible to represent the crack tip asymptotic displacement fields by using a local partition of unity method. However, the enrichment is only partial in the blending elements connecting the enriched zone with the rest of the mesh, and consequently pathological terms appear in the interpolation, which lead to increased error. In this study we propose enhancing the blending elements by adding hierarchical shape functions where appropriate; this permits compensating for the unwanted terms in the interpolation. This technique is an extension of the study of J. Chessa, H. Wang
and T. Belytschko
[Int. J. Numer. Methods Eng. 57, No. 7, 1015–1038 (2003; Zbl 1035.65122
)] to fracture mechanics problems. The numerical results show that the proposed enhancement always results in greater accuracy. Moreover, enhancing the blending elements makes it possible to recover the convergence rate that is decreased when the degrees of freedom gathering technique is used to improve the condition number of the stiffness matrix.