Multi-scale based stochastic damage evolution

In the present work, the stochastic damage evolution is developed by the random extension of the homogenization based multi-scale damage model. The energy bridging theorem is introduced as the bridging vehicle between micro and macro scales. And then the probability density evolution method (PDEM) and the generalized density evolution equation (GDEE) are introduced to describe the nonlinear transportation of randomness between scales. It is observed from the numerical results that not only the randomness induced stress field oscillation but also the stochastic damage evolution are well reproduced by the proposed framework.