  We propose a new class of phase field models coupled to viscoelasticity with
large deformations, obtained from a diffuse interface mixture model composed by
a phase with elastic properties and a liquid phase. The model is formulated in
the Eulerian configuration and it is derived by imposing the mass balance for
the mixture components and the momentum balance that comes from a generalized
form of the principle of virtual powers. The latter considers the presence of a
system of microforces and microstresses associated to the microscopic
interactions between the mixture's constituents together with a system of
macroforces and macrostresses associated to their viscoelastic behavior, taking
into account also the friction between the phases. The free energy density of
the system is given as the sum of a Cahn-Hilliard term and an elastic
polyconvex term, with a coupling between the phase field variable and the
elastic deformation gradient in the elastic contribution. General constitutive
assumptions complying with a mechanical version of the second law of
thermodynamics in isothermal situations are taken. We study the global
existence of a weak solution for a simplified and regularized version of the
general model, which considers an incompressible elastic free energy of
Neo-Hookean type with elastic coefficients depending on the phase field
variable. The regularization is properly designed to deal with the coupling
between the phase field variable and the elastic deformation gradient in the
elastic energy density. The analysis is made both in two and three space
dimensions.

Analysis of PDEs