A two-scale time-dependent damage model based on non-planar growth of micro-cracks

This paper presents the theoretical developments and the numerical applications of a time-dependent damage law. This law is deduced from considerations at the micro-scale where non-planar growth of micro-cracks, following a subcritical propagation criterion, is assumed. The orientation of the crack growth is governed by the maximum energy release rate at the crack tips and the introduction of an equivalent straight crack. The passage from micro-scale to macro-scale is done through an asymptotic homogenization approach. The model is built in two steps. First, the effective coefficients are calculated at the micro-scale in finite periodical cells, with respect to the micro-cracks length and their orientation. Then, a subcritical damage law is developed in order to establish the evolution of damage. This damage law is obtained as a differential equation depending on the microscopic stress intensity factors, which are a priori calculated for different crack lengths and orientations. The developed model enables to reproduce not only the classical short-term stress–strain response of materials (in tension and compression) but also the long-term behavior encountering relaxation and creep effects. Numerical simulations show the ability of the developed model to reproduce this time-dependent damage response of materials.