| Summary: | The computational simulation of failure in solids poses many challenges. A proper understanding of how structures respond under loading, both before and past the peak load, is important for safe and economical constructions. This requires numerical models for failure which are both faithful to the physical reality and mathematically well founded. A serious computational issue is that of objectivity with respect to the spatial discretisation of a problem. This requires that upon refinement of the spatial discretisation of a problem, a unique, physically meaningful result is approached. One approach to ensure objectivity with respect to spatial discretisation when simulating failure in solids is to allow displacement discontinuities in the solution. In this work, different techniques, of varying complexity, are developed to simulate displacement discontinuities which are independent of the spatial discretisation using finite elements. The different techniques are then critically evaluated. The first model examined involves adding only the effect of a displacement discontinuity to a finite element as an incompatible strain mode. This allows a traction-separation relationship to be applied at an interface and can be implemented simply in a standard finite element code. It is however shown that this type of model can be cast in an equivalent continuum format, a form which is known to be sensitive to the spatial discretisation. The second approach developed involves the addition of the Heaviside function to the underlying finite element interpolation basis. This method is based on the partition of unity concept, and allows the Heaviside function to be added locally to a finite element mesh to simulate a propagating displacement discontinuity. The approach is formulated for geometrically linear, geometrically nonlinear, quasi-static and dynamic problems. It is shown to be completely independent of the spatial discretisation. The partition of unity-based model is used also to simulate failure using a regularised strain softening model. When a critical level of inelastic deformation is reached, a displacement discontinuity is inserted. This model is better suited to modelling the entire failure process than a continuum or discontinuous model alone. Through numerical examples, it is shown that the inclusion of a displacement discontinuity during the failure process can lead to a different failure mode than for a continuum-only model.
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