Nonlocal damage models are typically used to model failure of quasi-brittle materials. Due to brittleness, the choice of a particular model or set of parameters can have a crucial influence on the structural response. To assess this influence, it is essential to keep finite element discretization errors under control. If not, the effect of these errors on the result of a computation could be erroneously interpreted from a constitutive viewpoint. To ensure the quality of the FE solution, an adaptive strategy based on error estimation is proposed here. It is based on the combination of a residual-type error estimator and quadrilateral h-remeshing. Another important consequence of brittleness is that it leads to structural responses of the snap-through or snap-back type. This requires the use of arc-length control, with a definition of the arc parameter that accounts for the localized nature of quasi-brittle failure. All these aspects are discussed for two particular nonlocal damage models (Mazars and modified von Mises) and for two tests: the Brazilian tensile splitting test and the single-edge notched beam test. For the latter test, the capability of the Mazars model to capture the curved crack pattern observed in experiments – a topic of debate in the literature – is confirmed.

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