Over the past years, the non-local method has established itself as an effective remedy to the well-known pathological mesh dependency that inherently affects softening media. The non-local method incorporates an intrinsic length into the traditional continuum theory and therefore the size of the localising zone is resolved, attenuating the unwanted effects of spurious mesh dependency. However, despite many contributions that have successfully employed the non-local theory, it is still not clear how exactly should non-locality be formulated in the general sense. Aiming to answer the question of which non-local formulations effectively lead to mesh-insensitive results, we select in this article several constitutive variables to be non-local quantities by taking both Lemaitre and Gurson–Tvergaard–Needleman models as the base for the non-local enhancement. The resulting non-local constitutive models are employed in the numerical simulation of various specimens which are subjected to different values of stress triaxiality and third invariant at the fracture zone. The goal is to find which models present the best performance in the task of providing mesh-insensitive solutions for different stress states. The results show that strain-softening mesh dependency is stronger in plane strain than in the axisymmetric case. It is also found that the variables that regularise the solution in the axisymmetric case do not necessarily eliminate mesh sensitivity in the other cases. Furthermore, the results indicate that damage should be the preferred non-local variable in the case of implicit damage models. This result is in sharp contrast with the case of explicit damage models, for which it has already been shown in the literature that damage is a bad candidate for non-local variable.
