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

TL;DR

This paper extends the Cahn-Hilliard and Cahn-Larché models by incorporating a damage variable, and proves the existence of weak solutions for these coupled systems, capturing phase separation, elasticity, and damage effects.

Contribution

It introduces a new framework combining phase separation, elasticity, and damage modeling, and establishes the existence of weak solutions for this complex coupled system.

Findings

Existence of weak solutions for the coupled Cahn-Larché and damage system.

Framework captures damage effects influenced by local surroundings.

Mathematical proof of solution existence under rate-dependent damage evolution.

Abstract

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.