Progressive damage in discrete models and consequences on continuum modelling

In phenomenological damage models, damage is very often understood as a degradation of the elastic stiffness of the material. The unrealistic features of damage localisation as a result of strain softening are usually circumvented by the introduction of an internal length in the continuum description that scales the localisation process and controls the size of the damage/strain localisation zone. This paper focuses on the motivation for introducing such an internal length. It is based on the analysis of failure in lattice models presenting an initial disorder in strength. The simple model introduced here and studied numerically, should be amenable to a continuum description for large enough lattice sizes. In the lattice modelling, an internal length appears as a correlation length owing to spatial redistributions and interactions during the failure process. The variations of this length with the system size are studied numerically and theoretically with an original model based on percolation theory, which accounts for the spatial interactions. The analysis shows that the internal length increases with the damage of the system, and finally reaches a finite (lattice size independent) value at the peak load. A full statistical analysis of the local stress is provided and discussed in the formalism of multifractals, so as to extract the salient scaling features of the model.