Two-scale FE–FFT- and phase-field-based computational modeling of bulk microstructural evolution and macroscopic material behavior

The purpose of this work is the development of a two-scale phase-field-based computational model for coupled microstructure evolution and macroscopic mechanical material behavior. To this end, the mechanical behavior of the macroscopic continuum is based on a model for each of its material elements or points as a unit cell (UC) whose average properties determine those of the corresponding material point. The UC model itself is based on materially inhomogeneous elasticity, with material inhomogeneity resulting from microstructure evolution in the UC, driven in particular by stress relaxation. Since the focus of the current work is on computational aspects, microstructure evolution is restricted here for simplicity to bulk structural rearrangement (e.g., austenite–martensite transformation). Consequently, the corresponding phase fields are non-conservative. As well for simplicity, the current formulation is restricted to geometric linearity. The algorithmic formulation and numerical solution of the microscopic initial–boundary-value problem (IBVP) for mechanical equilibrium and relaxational dynamics in the UC is based on Green function and (fast) Fourier transform (FFT) methods. On the other hand, the algorithmic formulation and numerical solution of the macroscopic BVP for quasi-static macroscopic mechanical equilibrium is based on finite element (FE) methods. To demonstrate its capabilities, the current two-scale computational approach is applied in the last part of the work to the modeling and simulation of single- and polycrystalline UCs undergoing structural rearrangement from austenite into multiple martensite variants during loading of the macroscopic continuum.