Isogeometric analysis of the Cahn-Hilliard phase-field model coupled with strain gradient elasticity

Phase separation in solids governs microstructure evolution and, therefore, effective material properties, and is commonly modelled using phase-field approaches based on the Cahn-Hilliard equation. At small scales, size-dependent material behaviour arises that cannot be captured by classical continuum models, motivating strain gradient theories with intrinsic length scales. Despite these developments, the coupling between phase separation and gradient elasticity has remained largely unexplored beyond theoretical formulations. In the present contribution, we fill this gap by developing a numerical framework and performing a comprehensive parameter study of a phase-field model based on the Cahn-Hilliard equation coupled with strain gradient elasticity. Two weak formulations of the coupled problem are proposed, which, due to the higher-order nature of the governing equations, require C 3 or C 2 continuity for the displacement field and C 2 or C 1 continuity for the concentration field, respectively. The problems are solved using NURBS-based isogeometric analysis, which naturally provides the required higher-order continuity. We show that strain gradient effects can significantly influence the kinetics of phase separation, particularly the steady-state phase distribution under bending-type loading. The proposed model and implementation provide a robust framework for the analysis of coupled phase separation processes in materials where intrinsic length scales are relevant.