We present an efficient and reliable approach for the numerical modelling of failure with nonlocal damage models. The two major numerical challenges – the strongly nonlinear, highly localized and parameter-dependent structural response of quasi-brittle materials, and the interaction between non-adjacent finite element associated to nonlocality – are addressed in detail. Efficiency is achieved with a suitable combination of load-stepping control technique and nonlinear solver for equilibrium equations. Reliability of the numerical results is ensured by an h-adaptive strategy based on error estimation. We use a residual-type error estimator for nonlinear FE analysis based on local computations, which, at the same time, accounts for the nonlocality of damage model. The proposed approach is illustrated by means of three application examples: the three-point bending test, the single-edge notched beam and Brazilian test. In addition, we present a new nonlocal damage model based on nonlocal displacements. Its good qualitative behaviour and attractive numerical properties are discussed and illustrated by means of a uniaxial tension test.
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