Scale-Bridging Phase-Field Modeling of Microstructure Evolution by FE$^2$ Computational Homogenization

Phase-field models have become a standard tool for simulating complex microstructure evolution in materials, but their application to engineering-scale components is often hindered by prohibitive computational costs arising from the need to resolve fine-scale features. To address this challenge, we propose a consistent homogenization framework for phase-field theory. By enforcing a Hill-Mandel-type condition of micro-homogeneity formulated in terms of Gurtin's microforces, a well-posed boundary value problem is derived for the representative volume element (RVE), establishing rigorous micro-macro relations for both the order parameter and its gradient. The theory is implemented within a computational two-scale (FE$^2$) scheme and validated against direct numerical simulations. Two distinct examples are investigated: a minimal Allen-Cahn model and a mechanically-coupled model for stress-driven martensitic phase transformation. The results demonstrate that the proposed framework can reliably predict the spatial and temporal evolution of the macroscopically averaged fields with reasonable accuracy.