The first objective of the thesis is the review of the analysis of three-dimensional cracks starting from the premises of the LEFM. The importance of second-order terms of the  Williams' series expansion is considered for the correct description of the stress field in three-dimensional problems.  The  study of three-dimensional cracks is usually performed using an extension of two-dimensional concepts that cannot be done in a straightforward manner.  An analyisis of the effect of triaxiality is done in this thesis, concluding that  singular terms are not enough to achieve the desired description and, at least, constant terms of Williams' series expansion have to be considered.



The second point is an introduction to the XFEM for three-dimensional crack modelling, including some of the  developments carried out in the last years. The main goal of the thesis is the computation of the SIF for  generic three-dimensional cracks with curvature.  Some improvements  in the formulation are introduced and  numerical verification and convergence studies are performed. The SIF computation in cracks that exhibit curvature  is a challengig issue.  The  formulation of the interaction integral has been modified to improve efficiency and  the effect of curvature in the gradient has been also introduced. For the gradient  formulation the differential geometry theory has been applied using the \LS description. This proposal significantly improves the performance of the domain integrals for curved and non-planar crack studies.

The last point is the study and introduction of an enrichment to improve the description of the existing state close to the corner singularity (or free border singularity). In addition to a review of the state of the art,  the spherical behavior of the free border singularity has been considered using a set of functions known as spherical harmonics that constitute  a basis for the description of phenomena with spherical symmetry. An enrichment based on these functions has been introduced   in the formulation of XFEM to reproduce the local singularity effect.