Cahn–Hilliard phase field theory coupled to mechanics: Fundamentals, numerical implementation and application to topology optimization

The presented framework couples the Cahn–Hilliard phase field theory to continuum mechanics using a variational principle. All equations follow consistently from stationary of a rate potential and yield a physically sound homogenization. Static and kinematic compatibility at the material interfaces are naturally guaranteed. In order to enforce admissibility of the phase field parameter, nonlinear complementary conditions are considered and embedded into the algorithmic formulation. Eventually, the variationally consistent framework also features topology optimization automatically. In contrast to other approaches that start from the optimization problem, the present formulation starts from a more comprehensive energy potential. This perspective allows to explore the natural physical mechanisms that control the system’s compliance (e.g., interface evolution) and that drive maximum structural performance (changing the direction of the evolution equation with respect to the phase field parameter). Furthermore, this perspective efficiently couples the physical constraints (e.g., mass and momentum conservation). Energetically optimized microstructures and an optimized beam structure illustrate the applicability as well as the numerical performance of the elaborated framework. •Fully variational Cahn–Hilliard phase field theory coupled to mechanics.•Incorporation of physically sound homogenization methods.•Static and kinematic compatibility at the interface are fulfilled.•Boundedness of the phase field parameter is enforced by means of NCP functions.•Energetically optimized micro-structures and topology are naturally included.