Abstarct: The modeling of a discontinuous field with a standard finite element approximation presents unique challenges. The construction of an approximating space which is discontinuous across a given line or surface places strict restrictions on the finite element mesh. The simulation of an evolution of the discontinuity is in turn burdened by the requirement to remesh at each stage of the calculation. This work approaches the problem by locally enriching the standard element-baxsed approximation with discontinuous functions. The enriched basis is formed from a union of the set of nodal shape functions with a set of products of nodal shape functions and enrichment functions. The eXtended Finite Element Method (X-FEM) is used to simulate the crack growth without remeshing. In X-FEM, the standard FE approximation domain is enriched with special functions to help capture the challenging features of a problem. Enrichment functions may be discontinuous (to model discontinuities in the field), their derivatives can be discontinuous (to model kinks in the field), or they can be chosen to incorporate a known characteristic of the solution (such as the square root singularity of linear elastic fracture mechanics). Applications to linear elastic fracture mechanics (LEFM), single static crack as well as crack growth are presented and demonstrated with several numerical examples