Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes. (English) Zbl 1340.35167

Heinemann, Christian; Kraus, Christiane

Math. Bohem. 139, No. 2, 315-331 (2014).

Summary: This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field.
The analysis is performed on a time-dependent domain which characterizes the nondegenerated elastic material regions. We choose a notion of weak solutions which consists of weak formulations of the Cahn-Hilliard system and the momentum balance equation, a variational inequality for the damage evolution and an energy inequality. For the introduced degenerating system, we prove global-in-time existence of weak solutions. The main results are sketched from our recent paper [WIAS preprint no. 1759 (2012)].

MSC:

35K65	Degenerate parabolic equations
35K55	Nonlinear parabolic equations
35J50	Variational methods for elliptic systems
74A45	Theories of fracture and damage
35K41	Higher-order parabolic systems

Keywords:

Cahn-Hilliard system; phase separation; complete damage; elliptic-parabolic degenerating system; linear elasticity; energetic solution; weak solution; doubly nonlinear differential inclusion; existence result; rate-dependent system