In this paper, we analytically investigate multi-component Cahn-Hilliard andAllen-Cahn systems which are coupled with elasticity and uni-directional damageprocesses. The free energy of the system is of the form$\int_\Omega\frac{1}{2}\mathbf\Gamma\nabla c:\nabla c+\frac{1}{2}|\nablaz|^2+W^\mathrm{ch}(c)+W^\mathrm{el}(e,c,z)\,\mathrm dx$ with a polynomial orlogarithmic chemical energy density $W^\mathrm{ch}$, an inhomogeneous elasticenergy density $W^\mathrm{el}$ and a quadratic structure of the gradient of thedamage variable $z$. For the corresponding elastic Cahn-Hilliard and Allen-Cahnsystems coupled with uni-directional damage processes, we present anappropriate notion of weak solutions and prove existence results based oncertain regularization methods and a higher integrability result for the strain$e$.
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机译：在本文中，我们分析性地研究了具有弹性和单向损伤过程的多组分Cahn-Hilliard和Allen-Cahn系统。系统的自由能的形式为$ \ int_ \ Omega \ frac {1} {2} \ mathbf \ Gamma \ nabla c：\ nabla c + \ frac {1} {2} | \ nablaz | ^ 2 + W ^ \ mathrm {ch}（c）+ W ^ \ mathrm {el}（e，c，z）\，\ mathrm dx $具有多项式对数化学能密度$ W ^ \ mathrm {ch} $，非均匀弹性能密度$ W ^ \ mathrm {el} $和损伤变量$ z $的梯度的二次结构。对于相应的弹性Cahn-Hilliard和Allen-Cahn系统，加上单向损伤过程，我们提出了弱解的适当概念，并基于某些正则化方法和较高的可积性结果证明了存在结果。
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