A time-dependent multi-scale damage model for rocks based on sub-critical growth of micro-cracks

The stability of underground cavities, such as the facilities for deep repositories of nuclear wastes, must be evaluated by considering the induced damaged zone due to the excavation phase, related to nucleation and growth of micro-cracks in the rock matrix. The evolution of the rock damage is time-dependent mainly because of sub-critical propagation of cracks. This paper presents the modeling of the macroscopic mechanical behavior of quasi-brittle rocks, using a micro-mechanical approach. The model is built in two steps. First, the elastic coefficients are calculated at the micro-scale in finite periodical cells, with respect to the micro-cracks length and their orientation. Then, these effective elastic properties are used in a sub-critical damage law in order to establish the evolution of the damaged zone. This damage law is obtained as a differential equation depending on time. The macroscopic behavior is determined, at each step, by an asymptotic homogenization procedure. The developed model enables to reproduce not only the classical short term stress-strain response of quasi-brittle rocks on complex loading paths (in tension and compression) but also the long-term behavior encountering relaxation and creep effects. In this model, the size of the microscopic cell, corresponding to the distance between two adjacent micro-cracks, is a material parameter that affects the macroscopic response of the material.