Existence results for diffuse interface models describing phase separation and damage

TL;DR

This paper proves the existence of weak solutions for complex coupled systems modeling phase separation and damage in materials, involving advanced mathematical techniques and regularization methods.

Contribution

It introduces a new framework for analyzing coupled Cahn-Hilliard and Allen-Cahn systems with elasticity and damage, establishing existence results under broad conditions.

Findings

Existence of weak solutions for the coupled systems.

Development of regularization techniques for analysis.

Higher integrability results for strain fields.

Abstract

In this paper, we analytically investigate multi-component Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form $\int_\Omega\frac{1}{2}\mathbf\Gamma\nabla c:\nabla c+\frac{1}{2}|\nabla z|^2+W^\mathrm{ch}(c)+W^\mathrm{el}(e,c,z)\,\mathrm dx$ with a polynomial or logarithmic chemical energy density $W^\mathrm{ch}$, an inhomogeneous elastic energy density $W^\mathrm{el}$ and a quadratic structure of the gradient of the damage variable $z$. For the corresponding elastic Cahn-Hilliard and Allen-Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for the strain $e$.