In this paper an extended finite-element method (X-FEM) that is fully compatible with standard FE program has been formulated based on a virtual node technique. A cohesive crack model that is appropriate for concrete fracture under mixed-mode loading has been integrated into the formulation. The proposed method was implemented into a commercial FE program as a user subroutine, and two benchmark experimental tests were successfully modelled. The numerical robustness and predictive power of the proposed method have been demonstrated by its excellent predictions on arbitrary crack evolution and the associated load–displacement curves. Detailed numerical investigation of the crack wake shielding effects on the fracture loading curves showed that: (a) crack wake shear shielding has little effect on the peak fracture load and its immediate neighbouring softening phase; (b) the initial shear cohesive stiffness has a significant influence on the descending slope of the softening part of the load–CMSD curve; and (c) the shear cohesive strength appears to make the softening phase more stable and to delay the abrupt fracture point.

In this paper, the extended finite element method (XFEM) is first applied to account for the weak discontinuity of the axisymmetric electrostatic field. Firstly, the interface between two materials in an element is described by the level set method. The enrichment function is used to modify the shape function of enrichment elements. Secondly, to illustrate the feature of the enrichment function, the distribution diagrams of enrichment functions in sub-elements are drawn. The 3D field can be simplified to an axisymmetric field, which can reduce the difficulty of calculation. Finally, models with bubbles in liquid nitrogen in the axisymmetric field are used to prove the reliability of XFEM. Compared with the conventional finite element method (CFEM), XFEM costs lower computing resources with almost the same computational accuracy.

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