Localization analysis of nonlocal models with damage-dependent nonlocal interaction

Abstract

This paper systematically evaluates (in the one-dimensional setting) the performance of a new type of integral nonlocal averaging scheme, initially motivated by the idea of internal time that reflects the reduction of the elastic wave speed in a damaged material. The formulation dealing with internal time is replaced by the equivalent concept of a modified spatial metric leading to a damage-dependent interaction distance. This modification has a favorable effect on the evolution of the active part of damage zone and leads to its gradual shrinking, which naturally describes the transition from a thin process zone to a fully localized crack. However, when a pure damage model (with no permanent strain) is considered, the resulting load-displacement diagrams exhibit dramatic snapbacks and excessively brittle behavior in the final stages of failure. The concept of damage-dependent interaction distances is therefore extended to damage-plastic models and damage models with inelastic (permanent) strain. It is shown that, for formulations that consider a part of the strain as irreversible, the overall stress-displacement response becomes realistic for quasi-brittle materials such as concrete, for which the diagram typically exhibits a long tail.