A Cahn-Hilliard model coupled to viscoelasticity with large deformations

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.

Visco-elastodynamics at large strains Eulerian

“…2. Let us note that the velocity here never exceed the wave speed C/ϱ of the fully inviscid nondispersive model (1) which was normalized to 1 in Fig. 2.…”

“…here S = φ ′ (F el )−div(ℓ 2 1 C∇F el ). This corresponds to the enhancement of the stored energy φ(F el ) by the term 1 2 ℓ 2 1 C|∇F el | 2 . Yet, the rigorous analysis seems, however, again analytically open, cf.…”

“…[33,Remark 9.4.6] or [51, Remark 2.3]. Here a combination with the dissipative gradient like in Sections 3.4 but of a higher order can help: with ζ in (73b) expanded by the 2nd-order gradient 1 2 Ω |ℓ 2 ∇ 2 ∂ ∂t F in | 2 dx which, when the derivative of ζ in (73b) understood in a functional sense, contributes to the left-hand side of (73b) by the 4th-order-gradient stress div 2 (ℓ 4 ∇ 2 ∂ ∂t F in ). For a rigorous analysis, we refer to [15].…”

“…One can also consider a conservative gradient as in the Poynting-Thomson standard solid in Remark 1, here enhancing S in in (73b) by the term −div(H∇F in ). In [50,51] and in [33, Sect.9.4], a simplified enhancement by 1 2 ℓ 2 C|∇F| 2 , leading to the higher-order gradient contribution to the Piola-Kirchhoff stress −Fdiv(ℓ 2 C∇F), was rigorously analyzed but, mixing F el and F in the stored energy is rather a misconception for a Maxwellian-type rheologies, corresponding rather to Zener's standard solid model.…”

Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentials

“…To cope with the case of singular potential, some authors (see [6]) suggested to include suitable relaxations. Besides, we refer to [19,20,28] and the references therein, for related nonlocal versions, to [1,17,24] for the additional coupling with elasticity, and to [27] for the coupling with the Keller-Segel equation.…”