A Cahn-Hilliard model coupled to viscoelasticity with large deformations

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.

Keywords

viscoelasticity, large elastic deformations, existence of weak solutions

2010 Mathematics Subject Classification

35Q35, 35Q74, 74B20, 74F10, 74H20

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This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics “F. Casorati”, University of Pavia.

In addition, A.A., P.C. and E.R. gratefully mention some other support from the MIUR-PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)” and their affiliation to the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (IstitutoNazionale di Alta Matematica).

Received 17 June 2022

Accepted 6 February 2023

Published 15 November 2023