A Cahn-Hilliard phase field model coupled to an Allen-Cahn model of viscoelasticity at large strains

TL;DR

This paper introduces a novel coupled Cahn-Hilliard and Allen-Cahn phase field model for viscoelastic materials at large strains, proving global existence of solutions and demonstrating stable finite element approximations with numerical tests.

Contribution

It develops a new regularization approach enabling analysis of more general, three-dimensional viscoelastic phase field models with nonlinear elastic energies.

Findings

Global existence of weak solutions in 3D for complex elastic energies

Development of unconditionally energy stable finite element schemes

Numerical validation with shape memory alloy simulations

Abstract

We propose a new Cahn-Hilliard phase field model coupled to incompressible viscoelasticity at large strains, obtained from a diffuse interface mixture model and formulated in the Eulerian configuration. A new kind of diffusive regularization, of Allen-Cahn type, is introduced in the transport equation for the deformation gradient, together with a regularizing interface term depending on the gradient of the deformation gradient in the free energy of the system. We study the global existence of a weak solution for the model. While standard diffusive regularizations of the transport equation for the deformation gradient presented in literature allows the existence study only for simplified cases, i.e. in two space dimensions and for convex elastic free energy densities of Neo-Hookean type which are independent from the phase field variable, the present regularization allows to study more general cases. In particular, we obtain the global existence of a weak solution in three space dimensions and for generic nonlinear elastic energy densities with polynomial growth. Our analysis considers elastic free energy densities which depend on the phase field variable and which can possibly degenerate for some values of the phase field variable. By means of an iterative argument based on elliptic regularity bootstrap steps, we find the maximum allowed polynomial growths of the Cahn-Hilliard potential and the elastic energy density which guarantee the existence of a solution in three space dimensions. We propose two unconditionally energy stable finite element approximations of the model, based on convex splitting ideas and on the use of a scalar auxiliary variable, proving the existence and stability of discrete solutions. We finally report numerical results for different test cases with shape memory alloy type free energy with pure phases characterized by different elastic properties.