Multi-phase-field method for heterogeneous brittle material with reduced-order-homogenization

The prediction of damage behaviors exhibited by heterogeneous materials within the phase field framework poses significant challenges, primarily owing to the highly intricate microstructure and different failure patterns. Although numerous models have been proposed to simulate crack propagation in composite materials, they are inclined to perceive damage as a solitary event and treat the composites as materials with homogeneity. However, this simplification might neglect the co-existence of different failure mechanisms at the microscopic level. In this paper, a multiscale multi-phase-field model is proposed to model damage propagation in heterogeneous brittle materials. The constitutive modeling is determined by performing simulations on the unit cell and using numerical homogenization. This multiscale model enables us to identify the damage mechanisms at the micro-scale while predicting the macro-scale response simultaneously. Moreover, in contrast to many multi-phase-field models that focus on two-phase composite materials, the proposed model exhibits the capacity to handle an arbitrary number of phase fields. To achieve this objective, each phase within the heterogeneous materials is assigned its own independent phase field. At the micro-scale, the damage of different phases is characterized by their respective individual phase fields. Then, the overall macro-scale mechanical response is determined by averaging and homogenization methods through the reduced-order-homogenization framework. By using multiple phase fields, we can track the damage evolution of different partitions by checking the corresponding phase field. The governing equations of the displacement field and each phase field are established based on the Francfort-Marigo variational principle. Additionally, by extracting the weak form of the governing equations, we implement this model within the finite element framework. Finally, this model is verified and validated through several sets of numerical examples, demonstrating that the damage evolution predicted by the proposed model is in good accordance with the references.