A Consistent Homogenization Framework for Phase-Field Theory Based on Gurtin's Microforce Balance

Phase-field models are a powerful tool for simulating microstructure evolution [1], but their microscopic resolution makes the direct simulation of macroscopic components computationally prohibitive. While two-scale homogenization methods like FE${}^2$ are well-established for mechanical problems, a rigorous framework for homogenizing the phase field and its gradient remains an open challenge. Current approaches are often limited to mechanics-only scale coupling or simple mixture rules that neglect gradient energy effects [2]. Even recent mathematically rigorous attempts, such as those using asymptotic expansion for fracture, are restricted in their scope [3], highlighting the need for a more general framework. This work presents a thermodynamically consistent homogenization framework for coupled phase-field problems. Building upon our prior work on phase-morphology kinematics [4], we establish a rigorous kinetic scale transition rooted in Gurtin's microforce theory [5]. Via a formal two-scale expansion of the order parameter, we derive a consistent transfer of the phase field and its gradient from the micro- to the macroscale. This yields a complete set of macroscopic constitutive laws and a macroscopic evolution equation for the homogenized order parameter. The framework's predictive capabilities are demonstrated within a computational two-scale simulation. The implementation leverages a micromorphic DirectFE${}^2$ scheme, previously established by the authors [6], to solve macroscopic boundary value problems for which the material response is directly governed by the evolution of the underlying microstructure.

References

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