Variationally Consistent Framework for Finite-Strain Microelasticity

Modeling microstructural evolution at large strains requires mechanical formulations that remain thermodynamically consistent while capturing significant lattice rotations and transformation-induced stresses. However, most existing finite-strain microelasticity and phase-field approaches apply macroscopic boundary conditions heuristically, preventing proper stress relaxation and violating the Hill-Mandel work equivalence required for homogenization. These limitations can misrepresent stress states and transformation pathways under finite strains. Here a variationally consistent finite-strain microelasticity framework is presented that couples microscopic and macroscopic mechanical equilibrium through a single energy functional. The resulting Euler-Lagrange conditions, periodic micro-equilibrium and macroscopic stress balance, are solved using a staggered FFT-Newton algorithm that combines a spectral fixed-point update for local fields with a Newton step for the homogenized deformation gradient. The formulation accommodates general hyperelastic constitutive laws and arbitrary transformation gradients. Benchmarks demonstrate accurate recovery of small-strain Eshelby solutions and systematic nonlinear deviations at large dilatations. Applied to deformation twinning in magnesium, the framework reproduces lenticular morphology, stress redistribution, and faster lateral growth consistent with experiments. This approach establishes a rigorous and scalable foundation for finite-strain phase-field simulations of coherent transformations under general stress or mixed boundary conditions.