A Cahn-Hilliard model coupled to viscoelasticity with large deformations

TL;DR

This paper introduces a novel phase field model coupling Cahn-Hilliard dynamics with viscoelasticity under large deformations, incorporating microforces, macroforces, and thermodynamic consistency, with proven existence of weak solutions.

Contribution

It develops a new coupled model in Eulerian coordinates combining phase separation and viscoelasticity with large deformations, including a proof of global weak solutions.

Findings

Model formulated in Eulerian configuration with micro and macro forces

Proved existence of weak solutions in 2D and 3D

Incorporates thermodynamic principles and elastic energy coupling

Abstract

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.