AbstractA methodology for the simulation of quasi‐static cohesive crack propagation in quasi‐brittle materials is presented. In the framework of the recently proposed extended finite element method, the partition of unity property of nodal shape functions has been exploited to introduce a higher‐order displacement discontinuity in a standard finite element model. In this way, a cubic displacement discontinuity, able to reproduce the typical cusp‐like shape of the process zone at the tip of a cohesive crack, is allowed to propagate without any need to modify the background finite element mesh. The effectiveness of the proposed method has been assessed by simulating mode‐I and mixed‐mode experimental tests. Copyright © 2003 John Wiley & Sons, Ltd.