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

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.

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. |