Probabilistic Formulation of Damage‐Evolution Law of Cementitious Composites

A model is proposed with a probabilistic evolution of damage in the material. The constitutive law is built through the development of a two‐level approach. The micro‐level is assumed to be constituted of elastic‐brittle springs whose strength follows a probabilistic distribution. The representative macro‐volume of material contains a given number of these elementary defects and its damage (loss of elastic properties) is computed from the knowledge of the local states. The macro‐behavior results from the interactions between all the micro‐defects. The model may be considered as representative of the behavior of brittle and almost brittle materials. It exhibits scattering and size effect. A Weibull distribution law is assumed for the local probabilities of failure and a parallel loose bundle connects the micro‐defects. These simple hypotheses lead to analytical expressions of the probabilistic constitutive law. The approach developed appears as an intermediate model between continuous damage mechanics and probabilistic brittle fracture. The knowledge of a single parameter $N_t\text{,}$ number of defects in a given volume, provides the degree of ductility of the material.