Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. (English) Zbl 1253.35179

Heinemann, Christian; Kraus, Christiane

Adv. Math. Sci. Appl. 21, No. 2, 321-359 (2011).

Summary: 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-Larché 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-Larché 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-Larché systems coupled with rate-dependent damage processes, we establish a suitable notion of weak solutions and prove existence of weak solutions.

MSC:

35Q74	PDEs in connection with mechanics of deformable solids
35D30	Weak solutions to PDEs
74A45	Theories of fracture and damage
74C10	Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
80A22	Stefan problems, phase changes, etc.

Keywords:

phase field approach; Cahn-Larché system; internal damage variable; unidirectional rate-dependent evolution inclusion; elliptic-parabolic systems