A localization analysis of a non-uniform damage lattice in presence of strength gradient

The failure of a non-uniform axial damage chain under uniform tension is studied both with discrete damage mechanics (DDM) and continuum damage mechanics (CDM). It is shown that a micomechanics-based nonlocal CDM model may be built from a DDM formulation, that may include material heterogeneities. DDM is based on a microstructured model consisting in multiples elastic-damage springs, whose elastic yield threshold is variable and depends on the position along the chain. We aim to develop a nonlocal CDM model as a relevant continuous formulation of the lattice DDM system. To do this, we rely upon a continualisation procedure applied to the difference formulation of the lattice problem, which gives us a nonlocal propagating damage model. The boundary conditions of the nonlocal CDM problem are equivalent to a finite length damage cohesive law. Analytical and numerical results show a strong proximity of the discrete and enriched continuous approaches for this heterogeneous bar problem, as well as the effectiveness of the nonlocal damage model to capture the softening localization phenomenon in heterogeneous quasi-brittle fields.