SUMMARYA model which allows the introduction of displacements jumps to conventional ÿnite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so-called 'embedded discontinuity' models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid ÿnite element that can be used and displacement jumps are continuous across element boundaries. Using ÿnite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi-brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes.

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Classical continuum models, i.e. continuum models that do not incorporate an internal length scale, suffer from excessive mesh dependence when strain-softening models are used in numerical analyses and cannot reproduce the size effect commonly observed in quasi-brittle failure. In this contribution three different approaches will be scrutinized which may be used to remedy these two intimately related deficiencies of the classical theory, namely (i) the addition of higher-order deformation gradients, (ii) the use of micropolar continuum models, and (iii) the addition of rate dependence. By means of a number of numerical simulations it will be investigated under which conditions these enriched continuum theories permit localization of deformation without losing ellipticity for static problems and hyperbolicity for dynamic problems. For the latter class of problems the crucial role of dispersion in wave propagation in strain-softening media will also be highlighted.

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Abstract-A continuum model that incorporates a dependence upon the Laplacian of the inelastic strain is used to regularize the initial value problem that results from the introduction of strain softening or non-associated flow. It is shown that the introduction of this gradient dependence preserves well-posedness of the initial value problem and that wave propagation in the enhanced continuum is dispersive. An analysis of the dispersive wave propagation reveals the existence of an internal length scale. Numerical analyses of one-dimensional and two-dimensional problems confirm that this internal length scale sets the localization zone and show that the results are insensitive to the fineness of the discretization and to the direction of the grid lines. This holds true with respect to the strain profiles, the energy dissipation and the extent of wave reflection.