Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage

A typical phase field approach for describing phase separation and coarsening phenomena in alloys is the Cahn-Hilliard model. This model has been generalized to the so-called Cahn-Larch\'e system by combining it with elasticity to capture non-neglecting deformation phenomena, which occur during phase separation and coarsening processes in the material. In order to account for damage effects, we extend the existing framework of Cahn-Hilliard and Cahn-Larch\'e systems by incorporating an internal damage variable of local character. This damage variable allows to model the effect that damage of a material point is influenced by its local surrounding. The damage process is described by a unidirectional rate-dependent evolution inclusion for the internal variable. For the introduced Cahn-Larch\'e systems coupled with rate-dependent damage processes, we establish a suitable notion of weak solutions and prove existence of weak solutions.