ABSTRACT

The aim of the present paper is to provide an overview of the main types of regularized continuum damage formulations that act as localization limiters, and to compare their localization properties by looking at the solutions constructed for a one-dimensional bar subjected to tension. Such a simple setting permits to demonstrate the principal concepts used by individual regularization techniques and to construct localized solutions based on relatively simple nonlinear ordinary differential (or integral) equations. Attention is paid to the onset of localization, seen as a bifurcation from a uniform state, as well as to the subsequent development of the damage profile up to complete failure. For each model, evolution of the size of active part of damage zone is documented and an equivalent cohesive diagram reflecting the relation between the stress and the inelastic part of bar elongation is constructed.

Inspection of the gallery of regularized damage formulations starts from integral-type nonlocal formulations, which incorporate weighted spatial averages of certain internal variables, and continues to gradient-enriched models in their explicit or implicit formats. Refined formulations with variable interactions are covered. Then, the focus shifts to regularization techniques that were initially developed outside the concrete mechanics community but recently have become widely popular. These formulations include variational damage models and phase-field models. Finally, the thick level set approach is described and analyzed. The results help to identify which regularization techniques are suitable for quasibrittle materials such as concrete.