Existence results for diffuse interface models describing phase separation and damage

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 ${\int }_{\Omega }\frac{1}{2}\mathbf{\Gamma }\nabla c:\nabla c+\frac{1}{2}|\nabla z{|}^2+W^{\mathrm{ch}}(c)+W^{\mathrm{el}}(e,c,z)\mathrm{d}x$ 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 $e$.

References in corpus (1)

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