  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
$e$.
