Phase-field modeling of stressed grain growth in nanocrystalline metals

Grain boundary excess free volume (GBEFV) plays a crucial role in the physical and mechanical behavior of polycrystalline materials and is a key parameter in materials design and grain boundary engineering, especially in nanocrystalline metals. Among various theoretical and experimental approaches for quantifying GBEFV, high-resolution differential dilatometry has recently been used to measure GBEFV release during annealing-induced grain growth. In this study, we develop a thermodynamically consistent constitutive theory for phase-field modeling of stressed grain growth in nanocrystalline metals, accounting for grain boundary expansion due to GBEFV. The theory is implemented in the finite element package Abaqus/Standard through a user-defined material subroutine (UMAT), coupling the finite-difference-based phase-field method for grain growth with the finite element method for mechanical equilibrium in a staggered scheme. We apply the model to simulate differential dilatometry experiments on nanocrystalline copper during annealing, enabling calibration and validation of the framework in linking GBEFV release to experimentally measured sample-level length changes. Our simulations show that GBEFV release during grain growth generates residual stresses of a few MPa and residual strains on the order of 10⁻⁵, arising from shrinkage mismatches at grain boundaries. Although these values are well below copper's elastic limit, the residual strains are comparable to the experimental linear strain of the sample (∼3 × 10⁻⁵), defined as the ratio of length reduction to initial length. Beyond the novelty of the theoretical development and its implementation in commercial software, our results demonstrate for the first time the significance of residual strain in measuring GBEFV from differential dilatometry.