SUMMARYA methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial di erential equation loses hyperbolicity. The approach is limited to rate-independent materials, so that the transition occurs on a set of measure zero. The discrete discontinuity is treated by the extended ÿnite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model. A new method was developed for the case when the discontinuity ends within an element; it facilitates the modelling of crack tips that occur within an element in a dynamic setting. The method is applied to several dynamic crack growth problems including the branching of cracks.

SUMMARYA new vector level set method for modelling propagating cracks in the element-free Galerkin (EFG) method is presented. With this approach only nodal data are used to describe the crack; no geometrical entity is introduced for the crack trajectory, and no partial di erential equations need to be solved to update the level sets. The nodal description is updated as the crack propagates by geometric equations. The advantages of this approach, here introduced and analysed for the two-dimensional case, are particularly promising in three-dimensional applications, where the geometrical description and evolution of an arbitrary crack surface in a complex solid is very awkward. In addition, new methods for crack approximations in EFG are introduced, using a jump function accounting for the displacement discontinuity along the crack faces and the Westergard's solution enrichment near the crack tip. These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack and are naturally coupled to the level set crack representation.

A method for modelling arbitrary growth of dynamic cracks without remeshing is presented. The method is based on a local partition of unity. It is combined with level sets, so that the discontinuities can be represented entirely in terms of nodal data. This leads to a simple method with clean data structures that can easily be incorporated in general purpose software. Results for a mixed-mode dynamic fracture problem are given to demonstrate the method.

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