Strain gradient enhanced phase-field model for ferroelectric domain evolution

• Phase-field model with strain-gradient elasticity for ferroelectric domain switching. • New Q84 finite element enables C0-continuous discretization. • Non-linear strain effects in polycrystalline ceramics with random initial polarization. • Strong strain gradients at grain boundaries and at intragranular domain walls. • Larger material length scale suppresses switching and shifts ferroelectric domains. Ferroelectric materials are widely used in industries, and accurately predicting their response under external loading is essential for both research and applications. With the miniaturization trend driving the size of electronic components to micro- and nanoscale, strain gradient effects become increasingly important. This paper presents a phase-field material model and governing equations for ferroelectrics that account for strain gradients and derive the associated weak forms. To circumvent the C 1 -continuity requirement of gradient elasticity, we employ a mixed finite-element formulation and present the discretization of a newly developed Q84 element for phase-field modeling, together with its element residuals, tangent matrix, and pseudocode for a user element subroutine. The implementation is verified by reproducing known results for monocrystalline and honeycomb ferroelectric specimens and, for the first time, we demonstrate strain gradient effects in these idealized settings. We then simulate a polycrystalline microstructure and analyze the influence of strain gradient elasticity for both uniform and random initial polarization. The results highlight the interplay between higher-order (strain gradient) effects and non-linear electromechanical coupling.